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A theory is presented in order to describe the density, elasticity and dynamic viscoelasticity of paper and other fibrous sheet in terms of their structure. The network model on which the theory is based is almost identical with that employed by Nimura and Kido, but the mathematical treatments are different. On computing the number of segments (portion of fiber between two adjacent junctions), the effect of the thickness of constituent fibers is considered. This leads to a excellent agreement between theoretical and experimental results.The density and the modulus of elasticity of a sheet are respectiN ely proportional to those of the constituent fiber, and increase rapidly with increasing the number of junctions. The correlation between the modulus and the density of the sheet depend markedly upon the effect of the thickness and the distribution function of the inclination of segments to the plane of sheet.

One may say that prior to the introduction of the Fermi-Dirac statistics into the theory of metallic conduction and allied phenomena a general mathematical method of attack on the various problems had been developed which necessarily still forms the basis of the modern treatment; but nevertheless in most cases the older theory had little success in predicting the order of magnitude, and in some cases, even the qualitative features of the various effects. However, the ground had been well prepared, so that as soon as it was realized that the electrons in a metal did not really obey the Maxwell but the Fermi-Dirac statistics, the mere introduction of the latter distribution function in the place of the former in the classical equations proved sufficient to clear away many of the old difficulties. Since the appearance of Sommerfeld’s paper in 1928 the first order effects have received on the whole a satisfactory explanation. In the case of the second order effects, however— and it is with one of these that the present paper deals—there are still very considerable difficulties to be faced. The problem of the change of resistance of a metal in a magnetic field has been treated by Sommerfeld, making use of a method which was originally developed by Gans. The calculations follow closely the classical treatment of Lorentz in that the mean free path of an electron is introduced phenomenologically as a parameter to be determined from the known experimental value of the conductivity. In the classical theory one pictures the process as follows. The metal is regarded as having a regular three-dimensional lattice structure with the metallic ions situated at the lattice points. It is further supposed that there are a certain number of conduction electrons, which might well correspond with the valency electrons, and that the assembly of conduction electrons obeys the classical distribution law. When an electric field is applied in a given direction the electrons are accelerated and experience elastic collisions with the metallic ions. Finally an equilibrium state is reached in which the number of electrons entering a given velocity range in unit time is just equal to the number ejected by collisions, and the mathematical expression of this state takes the form of an integral equation which must be solved to find the change in the original distribution function due to the applied field. From the change in the distribution function the conductivity is calculated. In the semi-classical calculations of Sommerfeld the model is the same except that the Fermi-Dirac statistics are used instead of the Max-wellian. If one compares the value of the conductivity, thus obtained, with the experimental value, one obtains a mean free path which is about a hundred times greater than the lattice spacing. This large value is not very plausible on classical ideas; but is readily understandable on wave mechanical principles.

1. In two famous papers on the Brownian motion of grains suspended in a stationary uniform fluid (liquid or gaseous) Einstein obtained, inter alia , the distribution function for the displacements of the grains during any interval t = 0 to t = τ from their positions at time t = 0. The object of this paper is to determine the distribution function for the more general case of a non-uniform fluid. The non-uniformity may refer to temperature, composition, or any other property which affects the coefficient of diffusion (D) of the grains in the fluid. The distribution function is given in 8, where it is shown how its accuracy might be experimentally tested. It contains terms additional to the one given by Einstein for the uniform case; certain of these are definitely determined, but another important term contains a coefficient that cannot be evaluated by considerations of the kind used in this paper (which depend purely on the conservation of the number of grains), but requires more detailed physical analysis; it is surmised that this coefficient vanishes in the case of Brownian particles which are large compared with the mean free path of the surrounding molecules. The main results of the paper refer to the rate of diffusion of the grains due to the non-uniformity of the fluid (9), and to the equilibrium distribution of the grains (10, 11). It is found that if their density is the same as that of the fluid, so that there is no tendency for them to settle in the lower strata, their steady distribution when the fluid is non-uniform is such that the concentration n (or number per unit volume of the fluid) is inversely proportional to D; a solution is also given for the case when the densities are not equal.

This paper provides simple proofs of two stardard .esults for the stable GI/G/1 queue on the structure of the distribution functions of the stationary virtual waiting time and the stationary queue-length Our argum-nt is applicable to more general single server systems than the queue GI/G/1. STABLE GI/G/I1 QUEUE; STATIONARY VIRTUAL WAITING TIME; STATIONARY QUEUELENGTH; STATIONARY ACTUAL WAITING TIME The objective of this paper is to provide simple proofs of two well-known results for the stable GI/G/1 queue, which results give the distribution function of W* (stationary virtual waiting time) and the distribution of Q (stationary queue-length) in terms of the distribution function of W(stationary actual waiting time). Our approach does not require transform methods and is applicable to more general single server systems than the GI/G/1 queue. Let {V,, n 2 0} and {U,?1, n > 0} denote independent sequences of independent and identically distributed, real, non-negative random variables defined on the probability space (Q, Y, P). The variable VI, represents the service time of customer n, and U,+1 is the time between the arrivals of customers n and n + 1, n 2 0. Denote by F and B the distribution functions of Vo and U, , respectively, and by f and b the first moments of F and B. We assume B is non-lattice, that each off and b is strictly positive and finite, and that f W* and Q(t) Q, as n -+ oo and t oo, where ">" denotes weak convergence. If J is a proper distribution on the real line with J(O-) = 0 and finite first moment j > 0, denote by j the proper distribution with J(0-) = 0 and Received in revised form 4 December 1973. Research supported by N.S.F. Grant No. GK-39073. 849 This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 04:45:48 UTC All use subject to http://about.jstor.org/terms 850 AUSTIN J. LEMOINE J(x) = p-1f: [ -J(y)]dy. Then, if Y is a random variable having distribution J, E{Y A x} = pf(x) for x = 0, where u A v = min(u, v) for real numbers u and v. Let H denote the distribution function of W, K = H * P, where "*" denotes convolution, and p = f/b. Let Bo denote the distribution function assigning unit mass to the origin, and Bi the j-fold convolution of B with itself, j _ 1. With the above notation, there are the following basic results from queueing theory; see Takics (1963) or Cohen (1969). Theorem 1. P{W* 1, d = Vo 0+ + V,_I, t, = U1 + -+ U,, and S, = d,t,. Next, set n = inf{n: n > 1 and S, 0, so that the queueing system is aperiodic in the sense of (6) from [2]. Since U, has a non-lattice distribution so does the variable ?. And, if m = E{7t}, then E{jt} = mf and E{r} = mb on account of Wald's Lemma. Hence, by (9) and (22) from [2], for x > 0, (1) mH(x) = E I1wx) }X n=O and (2) mbP{W* < xj} = E 1(W(s)xdS We have S1(w(S)x)ds I 1 1 Fd,-xds o n=1 . , = I I, I_-t_ I _yldy = 1 lis,-i+v.,-sy)dylls-_,+v._,_dy n=1 0 (3) = [(x+ U -S_1V._)+ -_(x-S--1V,,-W)] n=l1 This content downloaded from 157.55.39.244 on Fri, 17 Jun 2016 04:45:48 UTC All use subject to http://about.jstor.org/terms On two stationary distributions for the stable GI/G/1 queue 851 n-I = [(-S,)+ (x-S,V,)+] +x-S,-(x-SoVo)+

SummaryThe Sparre Andersen model assumes that the interclaim times and the amounts of claims are independent random variables, the former identically distributed according to a distribution function K(t), t ≥ o, K(o) = o, the latter identically distributed according to a distribution function P(y) — ∞ < y < ∞ As is well known, the Poisson risk process corresponds to the particular case K(t) = 1 — eβt. In the present paper it is pointed out that another particular case, viz. K(t) = є(t — h), corresponding to a fixed (and thus — strictly speaking—nonrandom) mterclaim time, h, has interesting applications. Thus, the ruin problem considered by Giezendanner, Straub and Wettenschwiler in a paper to the 1972 International Congress of Actuaries in Oslo can be formulated by means of this particular case. The same can be said about the earlier model brought forward by Ammeter in his 1948 paper in Skandinavisk Aktuarietidskrift.About the contents of the paper the following further information may be given. The general Sparre Andersen model is first presented and then the ruin formulas are given for the case with a positive gross risk premium. Thereafter, a modified and more direct method for deriving certain necessary auxiliary functions is illustrated by examples including 1 a the Giezendanner—Straub—Wettenschwiler model. The rest of the paper contains a discussion from the point of view of the Sparre Andersen model of (1) the discrete (equidistant) inspection of a Poisson process for ruin, (11) the Ammeter model and analogous models, and (111) the Giezendanner—Straub—Wettenschwiler model.

In this paper, we deal with sequences of iterated overshoot distribution functions. Under certain norming conditions and under the assumption that these sequences are convergent, the limits are completely characterized. The paper can be considered as a continuation of the work by Harkness and Shantaram [4]. As has been indicated by the referee, Shantaram and Harkness recently published a continuation of their work (see [5]). The present paper, however, is more general than [5]. The attention of the authors was directed to this problem while studying the behaviour of market demand transmitted through a chain of stock points.

Papers that have been written on the use of the rank order statistic for the two-sample problem under the Lehmann alternatives have focused on the use of rank tests to determine whether there is a difference in the two populations. In particular, the test proposed by Savage (1956) is the rank test that is usually suggested to be used in this situation. In this paper a Bayesian analysis of the problem based on the rank order statistic is presented. Suppose that a random sample of m observations is obtained from distribution 1 and an independent random sample of n observations is obtained from distribution 2, and N = m+n. The rank order statistic will be denoted by z = (z1, ...,ZN), where zi is 0 if the ith smallest observation is from distribution 1 and zi is 1 otherwise (i -1, ..., N). Let F(x) and G(x) be the distribution functions of distributions 1 and 2, respectively, both assumed to be continuous. If G(x) is the Lehmann alternative 1{1 F(x)}k(k > 0), which holds, for example, if the population distributions are both exponential or both Weibull, then the rank order probability is

In [8] the problem of nonparametric estimation in Markov processes has been considered, and estimates of the initial, two-dimensional joint, and transition densities of the process, satisfying a number of optimal properties, have been obtained. In the present paper, under the same nonparametric setup, the attention is centered primarily on the transition distribution function of the process. It will be assumed that the underlying Markov process, defined on a probability space $(\Omega, \mathscr{a}, P)$ and taking values in the real line $R$, is (strictly) stationary, and has initial, two-dimensional joint, and transition densities $p(\cdot), q(\cdot, \cdot)$, and $t(\cdot \mid x), x \varepsilon R$, respectively, relative to the appropriate Lebesgue measures. Let $K$ be a probability density. On the basis of the first $n + 1$ random variables $X_j, j = 1, 2, \cdots, n + 1$ of the process, we define the random variables $p_n(x), x \varepsilon R$, and $q_n(y), y \varepsilon R \times R$ (by suppressing the random element $\omega$) by the following relations \begin{equation*} \tag{1.1} p_n(x) = (nh)^{-1} \sum^n_{j=1} K((x - X_j)h^{-1})\end{equation*} \begin{equation*} \tag{1.2} q_n(y) = q_n(x, x') = (nh)^{-1} \sum^n_{j=1} K((x - X_j)h^{-\frac{1}{2}})K((x' - X_{j+1})h^{-\frac{1}{2}}),\end{equation*} where $h = h(n)$ is a sequence of positive constants satisfying also some additional conditions. We further set \begin{equation*} \tag{1.3} t_n(x' \mid x) = q_n(x, x')/p_n(x).\end{equation*} Next, by means of $p_n(x)$ and $q_n(x, x')$, define the random variables $F_n(x) = \int^x_{-\infty} p_n(z) dz, \quad G_n(z \mid x) = \int^z_{-\infty} t_n(dx' \mid x).$ We finally let $F(\cdot)$ and $G(\cdot \mid x), x \varepsilon R$, be the initial and transition distribution functions of the process. Under suitable conditions on the function $K$, the sequence $\{h\}$, and the process, the main results of this paper are the following. The distribution function $F_n(x)$, as an estimate of $F(x)$, obeys the Glivenko-Cantelli theorem. This is Theorem 3.1. Turning now to the estimate $G_n(\cdot | x)$ of $G(\cdot | x)$, we have been able to establish in Theorem 3.2 that $\sup \{|G_n(z | x) - G(z | x)a|; z \varepsilon R\}$ converges to zero, as $n \rightarrow \infty$, but in the probability sense. This is true for all $x \varepsilon R$. In Section 4 it is assumed that the $r$th absolute moment of $X_1$ exists (for some $r = 1, 2, \cdots$), and the problem is that of gaining further information about $G(\cdot \mid x)$, by estimating its $k$th moment, to be denoted by $m(k; x)$, for $k = 1, 2, \cdots, r$. By letting the rather simple expression $m_n(k; x) = (nh)^{-1}p_n^{-1}(x) \sum^n_{j=1} X^k_{j+1}K((x - X_j)h^{-1})$ stand for an estimate of $m(k; x)$, it is shown that, as $n \rightarrow \infty, m_n(k; x) \rightarrow m(k; x)$ in probability, for $k = 1, 2, \cdots, r$ and $x \varepsilon R$. This is the content of Theorem 4.1. Finally in Section 5 we look into the problem of estimating the quantiles of $G(\cdot \mid x)$, and in connection with this, two results are derived. For some $p$ in the interval (0, 1), it is assumed that the $p$th quantile, $\xi(p, x)$, of $G(\cdot \mid x)$ is unique. By defining $\xi_n(p, x)$ as the smallest root of the equation $G_n(z \mid x) = p$ and using it as an estimate of $\xi(p, x)$, it is proved that, as $n \rightarrow \infty$, $\xi_n(p, x) \rightarrow \xi(p, x)\quad \text{in probability (Theorem 5.1)},$ and $(nh)^{\frac{1}{2}} \lbrack \xi_n(p, x) - \xi(p, x) \rbrack \rightarrow N(0, \tau^2(\xi, x))\quad \text{in law}.$ This is Theorem 5.2, and the variance $\tau^2(\xi, x)$ is explicitly given in that theorem.

In the previous paper the conductivity of non-polar crystals in the strong electrostatic field was computed by the kinettc-statistical method. The impact ionization process, however, was neglected there. In the present paper the e!fect of the tmpact ionization process is also considered and the theory is extended to the case, where the distributiOn function depends on the velocity and the space-coordinate of the particles. The theory is applied to " the electron multiplication " in the p-n junction of Si, which was observed by McKay and McAfee. In the previous paper/l subsequent.ly referred to as part I, the conductivity of non­ polar crystals in the strong electrostatic field was computed by the kinetic-statistical method. Although the simplified model of the spherical energy surface was used, the results were in cp.talitative agreement with the observation of Ryder and Shockley. In the previous treatment, however, the impact ionization process was completely neglected, because the applied field was not strong enough to bring forth the above effect appreciably. Recently McKay and McAfee2l have found the " Electron Multiplication " phenomenon in the p-n junction of Si and Ge crystals. The theoretical explanation of the phenomenon was also given by McKay3l and Wolff4l by applying the method developed in the field of the gas· discharge. Therefore, we shall extend our previous theory with the intention of treating the electron multiplication process along the general line of the quantum theory of solids. Since the rigorous treatment of the elementary processes in solids is very complicated, we must at present content ourselves with rather crude approximations. Thus, our theoretical results are only qualitative.

Consider a sequence of independent random variables $x_1, x_2, \cdots, x_k, \cdots$ with mean 0 and variance $\sigma^2_k$. Let $S_n = (x_1 + \cdots + x_n)/s_n$ where $s^2_n = \sigma^2_1 + \cdots + \sigma^2_n$. The classical forms of the central limit theorem state that, with certain assumptions, the distribution function $F_n(x)$ approaches the Gaussian distribution $\Phi(x) = \frac{1}{\sqrt{2\pi}} \int^x_{-\infty} e^{-u^{2/2}} du.$ Berry [1] and Essen [3] have studied the behavior of $M_n = \underset{-\infty < x < \infty}\sup |F_n(x) - \Phi(x)|$ and in their main theorems have obtained bounds on $M_n$ which involve the moments of $x_k$ through the third. More generally consider a system of random variables $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ such that for each $n$, the variables $x_{n1}, \cdots, x_{nk_n}$ are independent. Let $S_n = x_{n1} + \cdots + x_{nk_n}$ and again let $F_n(x)$ be the distribution function of $S_n$. From a well known theorem of Khintchine [5] it follows that if the random variables $x_{nk}$ are infinitesimal (i.e., $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P\{|x_{nk}| > \epsilon\} = 0$ for every $\epsilon > 0$) then the class of possible limiting distributions of $F_n(x)$ coincides with the class of infinitely divisible distributions. Let $F(x)$ be any infinitely divisible distribution function and let $M_n = \sup_{-\infty < x < \infty} |F_n(x) - F(x)|$. In this paper we obtain bounds on $M_n$ in the case where $F(x)$ and the $x_{nk}$ have finite second moments. It is shown that under necessary and sufficient conditions for $F_n(x)$ to approach $F(x)$, the bounds on $M_n$ obtained approach zero as $n$ becomes infinite. Throughout the paper, given the system $(x_{nk})$ we shall let $F_{nk}(x), \varphi_{nk}(t), \mu_{nk}$, and $\sigma^2_{nk}$ be the distribution function, characteristic function, mean, and variance respectively of $x_{nk}$, and $F_n(x), \varphi_n(t), \mu_n$, and $\sigma^2_n$ have the same meaning for the random variable $S_n$.

In recent years, problems related to the characterization of the normal distribution from the property of stochastic independence of linear functions of independent random variables have been investigated by various authors. The most general result in this direction is that obtained independently by Darmois [4] and Skitovich [14], who proved that if there exist two linear functions $$X = \sum^n_{j = 1} a_jx_j \text{and} Y = \sum^n_{j = 1} b_jx_j\quad\text{with} a_ib_j \neq 0$$ $(j = 1, 2, \cdots n)$, such that they are stochastically independent, where $x_1, x_2, \cdots, x_n$ are $n$ independently (but not necessarily identically) distributed proper random variables, then each $x_j$ is normally distributed. Their methods of proof are similar in nature, both being based on the use of characteristic functions, without any assumption about the existence of moments. The same theorem has been proved independently by Basu [1], under the assumption that the random variables are identically and independently distributed and have finite moments of all orders. This result is also obtained independently by Lukacs and King [11], under the assumption that the random variables are independently (but not necessarily identically) distributed, each having finite moments up to order $n$, and by Linnik [10], under the assumption that the random variables have only finite variances. The special case of this theorem for $n = 2$ was proved earlier independently by Kac [6], Gnedenko [5], and Darmois [3], without any assumption about the existence of moments. Thus we see that the problems on the consequences of stochastic independence of linear statistics have been exhaustively studied. Hence the question that naturally arises is whether similar investigations into the distribution laws of the random variables are possible under the assumption of a suitable type of stochastic dependence of the linear statistics. In this direction, the author [8] has derived a characterization of the stable law with finite expectation from the property of linearity of regression of one linear statistic on the other for the case $n = 2$. The author [7] has also obtained some characterizations of the normal distribution from the consequence of the linearity of the multiple regression of one random variable on several others, when the variables have the linear structural setup as in the case of the bi-factor theory of Spearman. For the formulation of the problem investigated in the present paper, we require a precise definition of the terms conditional distribution, linearity of regression, and homoscedasticity. Let $F(x, y)$ and $F_0(x)$ denote respectively the distribution function of the two-dimensional random variable $(x, y)$ and the marginal distribution function of $x$. Then we define the conditional distribution function of $y$ for fixed $x$ by $F_x(y)$ so that it satisfies the relation \begin{equation*}\tag{1.1}\int^x_{-\infty} F_\xi(y) dF_0(\xi) = F(x, y).\end{equation*} In the present investigation, the following assumptions on the distributions of the random variables will be made: Assumption 1. The conditional distribution of $y$ for fixed $x$ as defined in (1.1) is assumed to exist, wherever needed. Assumption 2. Each of the random variables concerned has a finite second moment. This assumption allows us to take derivatives of the characteristic functions of the corresponding random variables up to and including the second order. Assumption 3. Without any loss of generality in the proof, it is assumed that the expectation of each of the random variables concerned is equal to zero. The role of these assumptions is to ensure the existence of the expectation and the variance of the conditional distribution of $y$ for fixed $x$, which may be defined respectively as \begin{align*}\tag{1.2}E_x(y) &= \int^\infty_{-\infty} y dF_x(y), \\ \tag{1.3}V_x(y) &= \int^\infty_{-\infty} \{y - E_x(y)\}^2 dF_x(y) \\ &= S_x(y) - \{E_x(y)\}^2, \end{align*} where $$S_x(y) = \int^\infty_{-infty} y^2 dF_x(y).$$ In this case, the regression of $y$ on $x$ is said to be linear, if the relation \begin{equation*}\tag{1.4}E_x(y) = \beta x\end{equation*} is satisfied for all $x$, except for a set of probability measure zero, as the expectations of both the random variables $x$ and $y$ are already assumed to be zero. The constant $\beta$ in equation (1.4) is called the coefficient of regression of $y$ on $x$. Similarly the conditional distribution of $y$ for fixed $x$ is said to be homoscedastic, if the conditional variance $V_x(y)$ as defined in (1.3) is a constant $\sigma^2_0$ not involving $x$. Thus if the regression of $y$ and $x$ is linear and given by $\beta x$ and the conditional variance of $y$ for fixed $x$ is a constant $\sigma^2_0$, we have the relation \begin{equation*}\tag{1.5}S_x(y) = \sigma^2_0 + \beta^2x^2\end{equation*} to be satisfied for all $x$, in addition to the condition (1.4). For simplicity in notation, throughout the present paper we shall use the term the conditional distribution of $y$ for fixed $x$ is L.R.H. ($\beta, \sigma^2_0)$ meaning thereby that the regression of $y$ on $x$ is linear and given by $\beta x$ and that the conditional variance of $y$ for fixed $x$ is $\sigma^2_0$, being free of $x$, i.e., equivalent to the statement that both the relations (1.4) and (1.5) are simultaneously satisfied. In the following sections we shall derive some characterizations of the normal distribution from the property of linearity of regression and homoscedasticity of suitable linear statistics. The main theorem (Theorem 3.2) is given in Section 3. The proof of this theorem uses as a starting point a very simple set of necessary and sufficient conditions for the linearity of regression and homoscedasticity (Lemma 2.1) and finally a theorem of Linnik (Lemma 2.3) on an analytical extension of Cramer's theorem [2] on the normal law. Several important corollaries are deduced in the subsequent section.

The approach to the traffic-flow problem based on an integral differential equation of the Boltzmann type which has been considered by one of us (I P ) in a recent paper is further developed. The possibility of passing is explicitly introduced into the equation for the velocity distribution function. As in the previous paper, it is shown that at sufficiently high concentration a collective flow process must take place. In order to study more specifically the effects of one car on another, we define reduced n-car distribution functions giving the probability of finding a cluster of n cars all having the same velocity. We derive an equation for the evolution of this distribution function. Study of it yields some information as to the way traffic changes from relatively free flow to completely hindered, “condensed” flow.

In many risk theoretical questions a central problem is the numerical evaluation of a convolution integral and much effort has been devoted over the years to mathematical and computational aspects. The paper presented to this colloquium by O. Thorin shows the subject to be topical but the present note stems from a recent paper by H. L. Seal, “Simulation of the ruin potential of non-life insurance companies”, published in the Transactions of the Society of Actuaries, Volume XXI page 563.In this paper, amongst other topics, Seal has presented some simulations of ruin probabilities over a finite time interval. Some years ago (Journal Institute of Actuaries Students' Society, Volume 15, 1959). I pointed out that the form of ruin probability could be expressed as a successive product of values of a distribution function. To my knowledge no one has attempted to see if this product form was capable of development and the numerical values in Seal's paper prompted me to spend a little time on the problem. In the time I had available it has not been possible to do more than experiment, but the conclusions reached may be of interest to other workers in this field. They showed that calculation is feasible but laborious. However, the knowledge that it can be done may suggest methods of improving the techniques.Seal's first simulation example is the calculation of ruin probabilities when the distribution of the interval of time between the claims is negative exponential and the individual claim distribution is also negative exponential. Instead of following Seal's method we can determine the “gain per interval” and find that if λ is the security loading the frequency function for the gain z isi.e. a Laplace distribution.

In a previous paper, the authors formulated the theory of irreversible processes in a spatially inhomogeneous plasma on the basis of Bogoliubov's theory as extended by Guernsey and Uhlenbeck. The kinetic equation, in terms of g(κ, n, t), the Fourier transform of the spatially inhomogeneous part f(r, n, t) of the one-particle distribution function F(r, n, t), has been obtained to the first order in 4πe2, all orders in (4πe2/ν), and the first order in κ which is related to the spatial inhomogeneity. In the present work, the mathematical part of the previous paper, especially the expansion in various orders in κ, has been revised. The kinetic equation is given in (33), in which are exhibited: (a) the stream term; (b) the corrections to the stream term arising from the "collisions"; (c) the Vlasov term; (d) the corrections, in the zeroth and the first order in κ, to the Vlasov term depending not only on the "static" effective field but also on the divergence of the mean current, these corrections being momentum-dependent and time-irreversible in nature; (e) the main "collision integral" which is time-irreversible, in the zeroth and the first order in κ.

In this paper the consistency of the two-sample tests of Sukhatme [6], Ansari and Bradley [1], Siegel and Tukey [5], and Mood [3] is investigated. Each of these tests is a distribution-free analogue of the F-test for testing the equality of the variances of two normal distributions. If the two samples are taken from continuous distributions with the same median (say 0) and distribution functions F(x) and F(x/a) respectively, the hypothesis tested states that a = 1 and the two-sided tests are consistent for a ≠ 1. If, however, the samples are from continuous distributions with the same median 0 and arbitrary distribution functions F and G, little is known about their asymptotic properties. In this paper the conditions for consistency for this case are given; further it is investigated if the corresponding estimates of differences in dispersion satisfy the usual properties of measures of dispersion, namely 1) that X and − X have the same dispersion and 2) that, if X has a larger (or equal or smalle...

The objective of the present paper is to define a robust point estimator of the parameter $\beta$ in the model \begin{equation*} \tag{1.1} y_j = \alpha + g_j(\beta) + z_j,\quad j = 1, 2, \cdots, n,\end{equation*} where $\alpha$ and $\beta$ are unknown parameters, $g_1, g_2, \cdots, g_n$ are real-valued functions of real variable satisfying suitable conditions and $z_1, z_2, \cdots, z_n$ are independent identically distributed random variables having a distribution function belonging to a specified class. An important special case of (1.1) is the regression model obtained by taking $g_j(\beta) = \beta x_j, j = 1,2, \cdots, n$, where the $x$'s are known constants. Robust point estimators of $\beta$ in this case have already been given by Adichie [1], who followed the method of Hodges and Lehmann and by Brown and Mood [7], who considered the so-called "median" estimators. The estimator presented in this paper, which is also a Hodges-Lehmann type estimator, therefore, provides a third alternative for the regression model. In Section 2 of this paper, a robust point estimator for $\beta$ in the model (1.1) under some suitable regularity conditions on $g_j$ and $z_j$ is defined. In Section 3, a simple computational technique for the calculation of this estimator is given. A small sample property of the estimator is given in Section 4, and in Section 5, asymptotic normality is established under some regularity conditions. In Section 6, some special cases of the model are considered.

Although a number of papers about the structure theory or the control theory of sliver have been published by many authers, it seems difficult to say that both of them have enough supported to each other. One of main reasons might depend on great difficulties to determine the distribution of fiber ends in sliver which is considered as the definite representation of sliver structure. In this report some trials to make the structure theory relate with the control theory are shown using a model sliver which consists of fibers having even thickness and length.At the first part of this report, a way of determining the positions of fiber ends in sliver from the sliver thickness which can be easily measured is discussed. Considering the relation between the fiber end positions and the sliver thickness, it is shown that positions of fiber ends can not be detected from the sliver thickness except those which are located at intervals of fiber length. Therefore it is clearly impossible to remove the irregularity within the fiber length only with the imformation of sliver thickness. Then a new method is proposed. This is, measuring and comparing the sliver thickness before and after drafting, positions of fiber ends at intervals of 1/N times fiber length can be detected if drafting ratio used is expressed by a rational number N/M (>1), where M and N are integers. In this way, it becomes possible to control the very short range irregularity of sliver thickness theoretically. This method also makes posible to count up the number of fiber ends existing in each intervals which divide the sliver along the length.In this paper, the length L/N, where L is the whole length of sliver and N is the total number of fibers, is chosen as the elementary interval and the standard deviation which is calculated from the set of the number of fiber ends contained in each elementary interval is assumed as a representative index for the irregularity of sliver thickness. However the elementary interval is usually so narrow that n times one would be reasonably used for practical observation. Then the next problems are how to estimate the standard deviation for the elementary interval from data for the n times elementary interval and to predict the irregularity of sliver thickness after control which is performed with data obtained by the n times elementary interval. For this purpose a method based on the maximum entropy principle of imformation theory is proposed and, applying this method to a random sliver, the expected irregularity of thickness controlled by ideal draft mechanism is investigated. Main results are as follows1. The expected irregularity of thickness is less improved when the sliver is controlled with data obtained by larger interval. This tendency rapidly becomes dull and, by and by, sliver control based on data for a interval over the critical one becomes ineffective.2. When the draft ratio smaller than one is forbidden to use, the expected irregularity of thickness after control is less improved than the one in the case of unlimited draft ratio.But results for both case close to each other no matter draft ratio is limited or not, when draftratio is large enough.

nential life distributions are accepted as characterizing the phenomenon of no wear. The problem of finding a class of life distributions which would similarly reflect the phenomenon of wear-out has been under investigation for some time. In answer to this problem we introduce in this paper the class of IHRA (Increasing Hazard Rate Average) distributions and show that it has, among others, the following optimal properties: (i) it contains the limiting case of no wear, i.e., all exponential distributions, (ii) whenever components which have IHRA life distributions are put together into a coherent system, this system again has an IHRA life distribution, i.e., a system wears out when its components wear out, and (iii) the IHRA class is the smallest class with properties (i) and (ii). 1. Introduction. The nature of many physical devices is to wear out in time. In this paper we endeavor to determine how the wear-out process is reflected as a property of the corresponding life distributions. Wear-out is an intuitively suggestive concept, but its precise stochastic meaning is not immediately evident. However, as a beginning, we review the long accepted stochastic characterization of the phenomenon of no wear. Let T ? 0 be the failure time of a device. Let F(t) = PI T > t} be the complement of the usual distribution function, which we will call the "survival probability." The conditional survival probability for remaining life, given that the device has survived to age x, is Fx(t) = F(x + t)/F(x) if F(x) > 0, Fx(t) = 0 if F(x) = O. A device does not wear if regardless of age an unfailed device is like new, i.e. Fx(t) = Fo(t) for all x, t > 0. It follows that the class of no wear survival probabilities is the class of solutions of the functional equation

The present paper gives the formal statements and proofs of the results illustrated in [1]. In a series of papers ([2], [3], [4]) the present author has been developing the minimum distance method for obtaining strongly consistent estimators (i.e., estimators which converge with probability one). The method of the present paper is much superior, in simplicity and generality of application, to the methods used in the papers [2] and [4] cited above. Roughly speaking, the present paper can be summarized by saying that, in many stochastic structures where the distribution function (d.f.) depends continuously upon the parameters and d.f.'s of the chance variables in the structure, those parameters and d.f.'s which are identified (uniquely determined by the d.f. of the structure) can be strongly consistently estimated by the minimum distance method of the present paper. Since identification is obviously a necessary condition for estimation by any method, it follows that, in many actual statistical problems, identification implies estimatability by the method of the present paper. Thus problems of long standing like that of Section 5 below are easily solved. For this problem the whole canonical complex (Section 6 below; see [1]) has never, to the author's knowledge, been estimated by any other method. The directional parameter of the structure of Section 4 seems to be here estimated for the first time. As the identification problem is solved for additional structures it will be possible to apply the minimum distance method. The proofs in the present paper are of the simplest and most elementary sort. In Section 8 we treat a problem in estimation for nonparametric stochastic difference equations. Here the observed chance variables are not independent, but the minimum distance method is still applicable. The treatment is incomparably simpler than that of [4], where this and several other such problems are treated. The present method can be applied to the other problems as well. Application of the present method is routine in each problem as soon as the identification question is disposed of. In this respect it compares favorably with the method of [4], whose application was far from routine. As we have emphasized in [1], the present method can be applied with very many definitions of distance (this is also true of the earlier versions of the minimum distance method). The definition used in the present paper has the convenience of making a certain space conditionally compact and thus eliminating the need for certain circumlocutions. Since no reason is known at present for preferring one definition of distance to another we have adopted a convenient definition. It is a problem of great interest to decide which, if any, definition of distance yields estimators preferable in some sense. The definition of distance used in this paper was employed in [9]. As the problem is formulated in Section 2 below (see especially equation (2.1), the "observed" chance variables $\{X_i\}$ are known functions (right members of (2.1)) of the "unobservable" chance variables $\{Y_i\}$ and of the unknown constants $\{\theta_i\}$. In the problems treated in [3], [9], and [11], it is the distribution of the observed chance variables which is a known function of unobservable chance variables and of unknown constants, and not the observed chance variables themselves. However, the latter problems can easily be put in the same form as the former problems. Moreover, in the method described below the values of the observed chance variables are used only to estimate the distribution function of the observed chance variables (by means of the empiric distribution function). Consequently there is no difference whatever in the treatment of the problems by the minimum distance method, no matter how the problems are formulated. The unobservable chance variables $\{Y_i\}$ correspond to what in [11] are called "incidental parameters"; the unknown constants $\{\theta_i\}$ are called in [11] "structural parameters". In [9] there is a discussion of the fact that in some problems treated in the literature the incidental parameters are considered as constants and in other problems as chance variables. In contradistinction to the present paper [3] (in particular its Section 5) treats the incidental parameters as unknown constants. The fundamental idea of both papers is the same: The estimator is chosen to be such a function of the observed chance variables that the d.f. of the observed chance variables (when the estimator is put in place of the parameters and distributions being estimated) is "closest" to the empiric d.f. of the observed chance variables. The details of application are perhaps easier in the present paper; the problems are different and of interest per se.

The possible occurrence of plasma instabilities has increased the necessity of further investigation on plasma oscillations since the Second International Conference in Geneva. Though a number of paper has already been presented, there still remain lots of unsolved problems.In this paper detailed considerations were made against the dispersion equations of potential function and of velocity distribution function respectively.Several new dispersion relations were derived and tabulated each for amplified, continuous and damped plasma oscillations.

In a recent paper dealing with a model of a relaxed crack, the plastic relaxation round an isolated crack is represented by an array of dislocations coplanar with the crack itself. Similar models are used here to discuss the behaviour of an infinite sequence of relaxed cracks with a constant distance of vertical separation. When the body deforms in an anti-plane strain mode, exact solutions are obtained for the extent of spread of plasticity from each crack and the associated dislocation distribution function. For deformation under plane strain shear conditions the form of the governing singular integral equation does not allow exact solutions to be obtained; instead iterative solutions have been found for the case of widely spaced cracks. A comparison of the results for anti-plane strain and plane strain conditions reveals wide differences; the paper thus emphasizes the need for exercising caution in using anti-plane strain models, whose behaviour it is relatively simple to analyze, as a substitute for the more frequent plane strain conditions which are met with in practice. Finally some applications of the plane strain results to fracture at stress concentrations are briefly discussed.

In the paper entitled “Estimation of jumps, reliability and hazard rate” [V. K. Murthy, in Ann. Math. Stat . 36 , 1032–1040 (1968)], the author has treated the problem of estimating the jumps of the probability distribution function. In this paper, the author proposes the sum of the squares of the jumps as a criterion of departure from absolute continuity of the distribution function and then solves the problem of estimation based on a random sample. More specifically, let F(x) be a probability distribution function. Assuming the singular part to be identically zero, F(x) can be decomposed into F ( x ) = F 1 ( x ) + F 2 ( x ), where f 1 ( x ) is an everywhere continuous function and F 2 ( x ) is a pure step function with steps of magnitude, say, S ζ j at the points x = ζ j , j = 1, 2,…, ∞ and that finally both F 1 ( x ) and F 2 (x) are nondecreasing and are uniquely determined. Based on a random sample X 1 , X 2 ,…, X N of size N , a consistent, and asymptotically normal class of estimators are obtained for the distance Δ = S ζ j 2 . If the singular part is not identically zero, the results are good almost everywhere.

The application of the kinetic theory to the investigation of viscosity and other physical phenomena of a gas involves, in general, the determination of the distribution of the molecular velocities in the non-uniform state. This is unnecessary in the case of one special molecular model, successfully dealt with by Maxwell, but for more general models his treatment is inapplicable. It was shown by Boltzmann that the function which expresses the law of distribution of the molecular velocities must satisfy a certain integral equation, but no solution of the equation was discovered nor any further progress made until Enskog, in 1911, applied to it the method of solution by series, and thereby obtained the form of the function. Later the same writer, by a more elaborate treatment of the problem, succeeded in obtaining a solution applicable to the most general molecular model of a monatomic gas, and used it to calculate the numerical values of the coefficient of viscosity, heat conduction and of diffusion. About two years before the appearance of Enskog’s later work, a paper was published by Chapman, in which he obtained the form of the velocity distribution function, partly by using the fact that it must be an invariant. The complete solution of the function was found by applying Maxwell’s equation of transfer to certain odd and even functions of the molecular velocity. The application of the solution to the evaluation of the coefficients of viscosity, heat conduction, and so on, yielded the same results as those afterwards obtained by Enskog.

We will in this paper consider the risk process from the point of view of random walk in one dimension. The particle starts out at the origin. Each claim is equivalent to a step in the random walk. The length of the step is equal to the amount of the claim minus the amount of the premium which has been obtained since the preceding claim. If the difference is positive the particle advances to the right and if the difference is negative to the left. At distance U to the right from the origin there is a barrier. The problem is to find the distribution function of X, the time it takes the particle to cross the barrier for the first time.In most practical applications of risk theory U is large in comparison to the individual steps of the particle. We will in this paper assume that U is large in comparison to the individual steps and draw certain conclusions about the risk processes from this assumption.The individual steps of the particle have a certain distribution. The corresponding characteristic function is ϕ. For reasons which will be seen later we will consider ϕ to be a function of it = θ instead of t. This means thatThe mean value and the standard deviation of each step is equal to m and σ respectively. We now writeWe now define two random variables X and Y.X = time to cross the barrier for the first timeY = X σ2/U2.

1. All the results of this paper are based on a general principle (Theorem I) of an asymptotic character. We show that this theorem leads to results in branches of the theory of functions generally regarded as quite distinct from each other. All the results on Watson's problem, on the theory of quasianalyticity, on one hand, and many known results on the distribution of the singularities of a Dirichlet series anid the distribution of the values taken by its analytic continuation, on the other hand, may be considered as particular cases of this theorem. As a matter of fact many new results far more general in both branches are obtained. Having established the main theorem we show, by a certain number of lemmas, cases in which the hypotheses of this theorem are satisfied. We then establish theorems with the same conclusions, their hypotheses being however more specific and easier to handle than those of the main theorem. Theorem I can be considered as a generalization of the principle of Cauchy's theorem which gives estimates of the coefficients of a Taylor series, or more generally of a Dirichlet series, by the maximum of the absolute value of the corresponding function on a circle around the origin (or on a vertical line for a Dirichlet series). The idea is to estimate the coefficients of expressions EZ.,dme-Cm (n ?1) which represent a function only "asymptotically," this estimate being made, on the other hand, only by means of the maximum of the modulus of the function on circles situated in any part of the plane, the radii of these circles diminishing with the increase of the upper density of the sequence {*XA} (lim sup n/X\). The closeness of approach is given by the evaluation of certain integrals. 2. Throughout this paper {*AX} (1 O) of quantities X,, smaller than x will be called the distribution function, or shorter the distribution of the sequence {AX }; clearly N(x) = 0 for x

A considerable difference between the fracture stresses theoretically calculated and those actually observed is usually explained by the presence of minute flaws, the so-called Griffith cracks, in the material.Assuming that the fracture is caused by the stress concentration around Griffith cracks, a statistical theory of brittle fracture is given which could be applied to arbitrary stress field. In this theory, the distribution of strength of test piece is related to the distribution of shapes of Griffith cracks.In this paper, it is supposed that the probability density function of the shape of Griffith cracks isf(ξ0)=δkδ/ξ0(δ+1)·exp{-(k/ξ0)δ}where ξ0 is the index of the shape or flatness of Griffith cracks.Based on this probability density function of the shape of Griffith cracks and some assumption concerning the stress concentration around cracks, the size effects of the uniaxial tensile and compressive strengths are calculated and given in the equations which coincide with the formula derived from Weibull's distribution function, that is, St=Ct·k·s·n-1/mfor the uniaxial tensile strength of the test piece which contains n cracks, where Ct is a constant, and s is the theoretically calculated fracture stress.In the calculation of the uniaxial compressive strength, besides Griffith's theory, a modified Griffith theory by McClintock and Walsh is considered.As the result of these calculations, the relations between the parameter of the distribution function of the shape of Griffith cracks δ and Weibull's coefficient of uniformity m are given in graphs. And the relations between the parameter δ and the“brittleness index”(the ratio between the uniaxial compressive and tensile strength) is given in another graph.Using these graphs, the dispersions of the actually observed uniaxial compressive and tensile strengths of rocks are discussed.To conclude, the following suggestions are given:(1) The brittleness index is not always a constant, but is a function of the distribution of shapes of Griffith cracks.(2) Besides the“inherent”dispersion discussed in this paper, there may be some experimental errors. Based on the theoretical aspect given in this paper, it is possible to evaluate the difference of experimental errors between the uniaxial compressive and tensile strength.

Let X be a random variable with distribution function F. Define for $x \in E^1 $ and $\beta \in ( {0,1} )$ the function $S( {x;\beta ,F} )$ as the infimum of the points z satisfying\[F( {x + S( {x;\beta ,F} )} ) - F( {x - S( {x;\beta ,F} )} )\geqq \beta .\] In an earlier paper some of the properties of S were established. This paper examines other aspects of S. A relationship is derived between points of discontinuity of F and points of nondifferentiability of S. Under some weak conditions S is shown to be continuous even if F has discontinuities of the first kind. The function S is also related to quantiles and estimation in a parametric setting.

Consider random samples from two independent distributions with absolutely continuous distribution functions $F(z)$ and $F(z - \theta)$, respectively. For testing the hypotheses $\theta = 0$ against $\theta > 0$, Hodges and Lehmann [6] show that the Pitman asymptotic efficiency of $W$, the Mann-Whitney form of the Wilcoxon statistic, with respect to the $t$-statistic is never smaller than .864 and in their 4th Berkeley Symposium paper [7] they indicate this efficiency is almost always greater than or equal to 1 for distributions with tails at least as heavy as those of a normal distribution. Hence for distributions with heavier tails, $W$ is a more robust statistic than the $t$-statistic. For the moment, consider a single sample of size $n$ from a distribution with absolutely continuous distribution function $F(z - \theta)$. For distributions with heavier tails some authors have proposed the $\alpha$-trimmed mean as an estimate of $\theta$; that is, the mean based on the middle $n - 2\lbrack n\alpha\rbrack$ observations. Tukey [16] and Huber [8] study this statistic when $F$ is a contaminated normal distribution. Bickel [3] studies the asymptotic relative efficiency properties of the $\alpha$-trimmed mean relative to the mean for the class of continuous distributions with symmetric, unimodal densities. For estimating the location of a Cauchy distribution, Rothenberg, Fisher and Tilanus [12] show the trimmed mean based on the middle 24 percent of the observations is the most efficient trimmed mean relative to the maximum likelihood estimate. This single sample statistic suggests a Mann-Whitney statistic based on trimmed samples. It is hoped that the effects of contamination by gross errors in the underlying distributions can be considerably reduced by considering such a statistic. We have the benefits of using a simple and well known rank statistic and, at the same time, of being able to increase the efficiency by adjusting the trimming proportions according to the weight in the tails of the underlying distributions. This type of statistic is also related to rank tests for censored data which have been studied by Basu [2], Gastwirth [5] and Sobel [14], [15]. If we have samples size $m$ and $n$ from absolutely continuous distributions corresponding to $F(z)$ and $F(z - \theta)$, respectively, we denote by $W_\alpha$ the Mann-Whitney statistic based on the middle $m - 2\lbrack m\alpha\rbrack$ and $n - 2\lbrack n\alpha\rbrack$ observations of the samples. We refer to $W_\alpha$ as the $\alpha$-trimmed Mann-Whitney statistic. It is the purpose of this paper to investigate the Pitman asymptotic relative efficiency properties of $W_\alpha$ for a sub-class of absolutely continuous distributions. First some definitions are given in Section 2. In Section 3 and Section 4 we establish the asymptotic normality of $W_\alpha$, derive the efficiency of $W_\alpha$ relative to $W$ and give some examples. The results of Section 5 include a greatest lower bound on this efficiency. It is interesting to note that this bound is the same one found by Bickel [3] for the efficiency of the trimmed mean relative to the mean.

Let $Z = \{z_1, z_2, \cdots, z_m\}$ be a $p \times m$ random matrix where $z_i$ are independently distributed according to $p$-variate normal distributions with means $\mu_i$ and common covariance matrix $\Lambda = (\lambda_{ij})(> 0$, positive definite) and let $nS_n = n(s_{ij})$ be a $p \times p$ matrix which is independent of $Z$ and is subject to a central Wishart distribution $W_p(\Lambda, n)$ with $n$ degrees of freedom and covariance matrix $\Lambda$. Hotelling's generalized $T_o^2$-statistic is then defined by \begin{equation*}\tag{1.1} T_o^2 = \operatorname{tr} S_n^{-1}ZZ' = \sum^m_{i=1} z_i'S_n^{-1}z_i\end{equation*} which is the statistic proposed by Hotelling [4], [5] for testing the hypothesis $H: M = \{\mu_1, \cdots, \mu_m\} = 0$ against $K: M \neq 0$. The distribution of $T_0^2$ when $H$ is true has been treated by several authors: for example, Hotelling [5], Ito [6], Siotani [13], [14], Pillai and Samson [12], and Davis [3]. Even in the null case, the exact distribution of $T_0^2$ is not available except for certain special values of $p$ and $m$. When $m = 1, T_0^2$ reduces to Hotelling's generalization of "Student's" $t$ and if $p = 1, (1/m)T_0^2$ is simply $F$-statistic. Hence the non-null distributions in these cases are known exactly. For the general non-null case, Constantine [2] has obtained the exact distribution as well as moments of $T_0^2/n$ using the generalized Laguerre polynomials of matrix argument. Unfortunately this distribution is valid only over the range $0 \leqq T_0^2/n < n 1$ and hence not so useful since we are usually interested in the upper tail of the distribution. Siotani [15] has treated an asymptotic expansion for the non-null distribution of $T_0^2$ according to the basic idea due to Welch [17] and James [9]. The same problem has been attacked by Ito [7], using the integral representation of the characteristic function of $T_0^2$. However formulas of these authors are inadequate for a good approximation to the distribution since they have only the terms up to order $n^{-1}$ (for some numerical information, see [8]) and also somewhat inconvenient terms for numerical work. Khatri and Pillai have evaluated the moments of their statistic $U^{(p)}$ (a constant times $T_0^2$) and given approximate distributions of $U^{(p)}$ (and hence of $T_0^2$) in the light of the first four general noncentral moments, the summary of which can be obtained in their recent paper [10]. In this paper an asymptotic expansion for the non-null distribution of $T_0^2$ is given up to the terms of order $n^{-2}$, in which the effect of the noncentrality is contained in the form $s_j = \operatorname{tr} \Omega^j, j = 1, 2, \cdots$ where $\Omega = \Lambda^{-1}MM'$. This is carried out by expanding the characteristic function of $T_0^2$ along the Welch-James idea and by inverting the resultant into the corresponding expansion of the distribution function or the density function.

A method for determining the spatial distribution in density of a plasma has been outlined in a previous paper. The method involves measuring the dispersive microwave propagation coefficients simultaneously at several frequencies and comparing the measured values with the values calculated for various distribution functions. The present paper demonstrates experimental verification of the method and describes the procedure used to determine the distribution function in a sample laboratory plasma experiment.

We develop a theory for distribution functions which are defined on an abstract linearly ordered space. This extends and gives additional insight into the work of J. H. B. Kemperman who was concerned with a special class of linearly ordered spaces. Attention is given to nonparametric appli- cations. The utility of Kemperman's work on tolerance regions and of related applications to goodness of fit tests, appearing in this paper, is en- hanced. In particular, it is shown that a number of standard nonparametric procedures can be extended to cover almost any kind of random sample (e.g., multivariate and most time series data) occurring in practice. 1. Introduction and summary. The spaces studied by modern probabilists have become very general. Frequently some structure is imposed on a space, but typically not an ordering. Since distribution functions can only be defined on partially ordered spaces (classically, on finite dimensional Euclidean spaces), they have become the "poor cousins" of probability measures. Nevertheless, distribution functions serve several important roles: (a) For the probabilist, they concisely conserve the information represented by a probability measure on a finite dimensional Euclidean space. (b) For the statistician, among other things, they enter into the definition of certain goodness of fit test statistics such as the Kolmogorov statistic D,. In this paper we only consider distribution functions (df) for linearly ordered spaces. We present what we believe to be a reasonably complete theory for abstract linearly ordered spaces. An effort is made to find parallels with results for the real line. We find necessary and sufficient conditions for a df to uniquely determine the probability measure giving rise to it. The theory specializes for a class of spaces introduced by J. H. B. Kemperman (1956). He was studying the subject of nonparametric tolerance regions. We consider alternative applications-nonparametric goodness of fit tests-in a slightly more general setting than his. On the practical side, Theorem 3 below permits one to apply a wide range of nonparametric statistical procedures to many new situations. What is required is a random sample with the observations assuming values in a separable metric space. This includes multivariate data and many types of times series data.

In this paper, the asymptotic behavior of decision procedures will be studied for a particular class of multiple decision problems. The study will throw some light on the desirability of the minimax decision procedure when the number of observations is large, and it will be seen that decision procedures frequently exist which are superior to the minimax decision procedure for large samples. The fact that the minimax decision procedure may be desirable in certain problems for small samples but undesirable for large samples was revealed by Hodges and Lehmann [1] in connection with estimation problems. Robbins [2] suggested the term "asymptotically subminimax" for the type of superior procedures which may then exist. A definition of this term which will be useful for an investigation of the asymptotic behavior of decision procedures, will be given in Section 1. A major part of this paper will be concerned with certain sequences of decision procedures called asymptotically admissible which have desirable properties similar to those of admissible decision procedures for the case of some fixed sample size. These asymptotically admissible decision procedures include a subclass of the asymptotically subminimax procedures, and the sequences of minimax procedures for those problems for which asymptotically subminimax procedures do not exist. The problems to be considered are those in which a random variable, $X$, is known to have a distribution function belonging to the distribution space, $\Omega = \{F_i(x)\}, i = 1, 2, \cdots, k$, and it is desired to select the true distribution function based on a sample of $n$ independent observations of $X$. It will be assumed that all $F_i(x)$ are absolutely continuous distribution functions having density functions, $f_i(x)$, and that for every constant $K$, the set of points for which $f_i(x)/f_j(x) = K (i \neq j)$ is a set of probability measure zero under every $F$ and all possible $i$ and $j$. A simple loss function, $W(F_i, d_j)$, where $d_j$ is the decision to select $F_j$, will be used with $W(F_i, d_j) = 1$ if an incorrect decision is made (i.e. $i \neq j)$, and $W(F_i, d_j) = 0$ if a correct decision is made (i.e. $i = j$). For such a loss function, the expected loss is simply the probability of making an incorrect decision. Section 1 will be concerned with asymptotically minimax sequences of decision procedures and Section 2 will be concerned with asymptotic admissibility. It will be seen that the asymptotic behavior of the minimax decision procedure depends on the limits of the components in the sequence of least favorable a priori distributions. Theorem 2.2 gives a sufficient condition, in terms of these limit values, for the minimax procedure to be asymptotically admissible. In Section 3, a detailed study will be made of the class of problems where $\Omega$ consists of $k$ univariate normal distributions having the same variance but different means. Let the means be denoted by $\theta_i (i = 1, 2, \cdots, k)$ with $\theta_1 < \theta_2 < \cdots < \theta_k$, and let $\min_i(\theta_{i+1} - \theta_i) = \gamma$. Then Theorems 3.1 and 3.6 will show that the minimax procedure is asymptotically admissible when the means can be put into sets, each set containing the same number, $\bar{n} \geqq 2$, of consecutive means, with a difference of $\gamma$ between any two consecutive means of a set, and a difference greater than $\gamma$ between any two means not belonging to the same set. Theorems 3.2 and 3.3 will show that in all other cases the minimax procedure is asymptotically inadmissible, and asymptotically subminimax procedures will be constructed for all these cases. Although a complete study of the asymptotic admissibility of asymptotically subminimax procedures will not be made in this paper, Theorem 3.7 will show that a certain asymptotically subminimax procedure is asymptotically admissible for all the cases covered by Theorem 3.3 and for some of the cases covered by Theorem 3.2. On the other hand, for those cases covered by Theorem 3.2 with an $\Omega$ consisting of only 3 means, it will be shown that every asymptotically subminimax procedure is asymptotically inadmissible.

In this paper, a two-sample, two-stage nonparametric estimation problem will be studied. The parameter $\theta = \theta(F, G)$ under consideration is estimable (i.e., there exists an unbiased estimator $\phi = \phi(X_1, \cdots, X_r; Y_1, \cdots, Y_s)$ of $\theta$). $\phi$ is a function of independent observations from two populations with cumulative distribution functions $F(X)$ and $G(Y)$. The functions $F(X)$ and $G(Y)$ belong to a specified class $D$, such that a $U$-statistic based on $\phi$ is the unique minimum variance unbiased estimator of $\theta$. The total number of observations on populations $X$ and $Y$ will be a fixed number $N$. The sampling procedure is carried out in two stages. First, take $M$ observations from each of the populations; then allocate the remaining $N - 2M$ observations between the populations. The method of allocation utilizes the information from the first stage observations. Two kinds of two-stage estimators, represented by $U'$, and $U'$ will be introduced in this paper. Both $U'$ and $U"$ are $U$-statistics with random sample sizes. $U'$ is based essentially on the second stage observations only. $U''$ is defined on all $N$ observations. Intuitively, the statistic $U''$ is more appealing. The first stage observations are used not only to determine the allocation of the second stage observations, but also to estimate the parameter $\theta$. (see Section 3) One of the main results (Section 4) is that $U'$ is unbiased and under certain conditions, the variance of $U'$ approaches asymptotically to a particular variance $V_0$. (Here we shall consider the cases that both the variances of $U'$ and $U''$, are finite.) $U''$ is in general biased. However, under the same conditions the value $E(U'' - \theta)^2$ approaches asymptotically to the same value $V_0$. This value $V_0$ is the smallest variance of any one-stage $U$-statistic estimator of $\theta$, subject to the restriction that the total number of observations on $X$ and on $Y$ is $N. V_0$ is computed (see Section 2) when the best one-stage allocation of $N$ observations to the two populations is made with the help of partial or even complete information about the distributions $F(X)$ and $G(Y)$. Such information about $F$ and $G$ is represented by the "nuisance parameters" $b_{10} = b_{10}(F, G), b_{01} = b_{01}(F, G)$, etc., defined in Section 2. No prior knowledge of $b_{10}$ and $b_{01}$ is required to compute $\operatorname{Var}(U')$, and $E(U'' - \theta)^2$. In Section 5, the "optimal" choice of the first stage sample size $M$ relative to the fixed total sample size $N$ is discussed. The term "optimal" is in the sense that the particular choices of $M$ in relative order of magnitude of $N$, such that as $N$ goes to infinity, the ratios $\operatorname{Var}(U')/V_0$ and $E(U'' - \theta)^2/V_0$ approach unity as fast as possible in order of magnitude of $N$. Three cases with different conditions on $\phi$ are considered. It is found that the "optimal" choices depend on the specific conditions. Section 6 contains some examples. To each $\theta(F, G)$, the corresponding estimators for $b_{10}$ and $b_{01}$ together with their behavior under different conditions on $F$ and $G$, will be given. The examples include cases where the proposed procedures can be applied as well as cases where it cannot be applied. Section 7 shows the asymptotic normality of $U'$ and $U''$. Section 8 indicates that the proposed procedures can be extended to $k$-sample case, for $k > 2$, with similar results. The technique of two-stage estimation has been used in several papers. Stein [12] used it to determine confidence interval of a pre-assigned length for the mean of a normal population with unknown variance. Putter [8] used it to estimate the mean of a stratified normal population, Robbins [10] discussed such a technic for the design of experiments. Later, Ghurye and Robbins [4] used it to estimate the difference between the means of two normal populations (or some other specified populations). Richter [9] discussed the estimation of the common mean of two normal populations. During the preparation of the present paper, Alam [1] discussed the estimation of the common mean of $k \geqq 2$ normal populations. This paper generalizes these two-stage procedures in two ways. First, the underlying cumulative distributions $F, G$ are members of a larger class of distributions. Secondly, the underlying parameters $\theta(F, G)$ are not restricted to population means or functions of means. Consequently, in such a general setup the question of "the best" estimator of any particular parameter $\theta(F, G)$ is not considered in this paper.

This paper studies the problem of testing the independence of two random variables $X, Y$ from a random sample, $(X_1, Y_1), (X_2, Y_2) \cdots (X_n, Y_n)$, of size $n$, where $X$ and $Y$ are angular variates (i.e., reals modulo 1). In the standard case where $X$ and $Y$ are ordinary real variables, the following approach has been useful. Suppose $(X, Y)$ has the continuous distribution function $F(x, y)$ with marginal distribution functions $F_1(x)$ and $F_2(y)$, respectively. It is desired to test $H_0: F(x, y) = F_1(x)F_2(y)$ against the alternative $H_A: F(x, y) \neq F_1(x)F_2(y)$. Let $F_n(x, y)$ denote the sample distribution function of the random bivariate sample, i.e., if $H(x)$ denotes the left continuous Heaviside function then \begin{equation*}\tag{1}F_n(x, y) = \frac{1}{n} \sum^n_{j=1} H(x - X_j)H(y - Y_j).\end{equation*} Also, let $F_{n1}(x)$ and $F_{n2}(y)$ denote the sample distribution functions associated with the first and second components of the random sample vector. In terms of $H(x)$ \begin{equation*}\tag{2}F_{n1}(x) = \frac{1}{n} \sum^n_{j=1} H(x - X_j)\end{equation*} and \begin{equation*}\tag{3}F_{n2}(y) = \frac{1}{n} \sum^n_{j=1} H(y - Y_j).\end{equation*} Blum, Kiefer and Rosenblatt [1] studied the following distribution free tests of independence based on the sample distribution function. Reject for large values of \begin{equation*}\tag{4}A_n = \sup_{x,y}|T_n(x, y)|\end{equation*} or \begin{equation*}\tag{5}B_n = n \int\int\lbrack T_n(x, y)\rbrack^2 dF_n(x, y),\end{equation*} where \begin{equation*}\tag{6}T_n(x, y) = F_n(x, y) - F_{n1}(x)F_{n2}(y).\end{equation*} The first statistic, constructed in the spirit of Kolomorov-Smirnov statistics, has good power properties, cf. [1], Section 4, but its asymptotic distribution is unknown. The statistic $B_n$ is analogous to the Cramer-von Mises statistic and is also equivalent to a statistic originally proposed by Hoeffding [3]. The characteristic function of the null asymptotic distribution of $B_n$ is (cf. [1] and [3]) \begin{equation*}\tag{7}E e^{izB} = \prod^\infty_{j,k=1} \big(1 - \frac{2iz}{\pi^4j^2k^2}\big)^{-\frac{1}{2}}.\end{equation*} Asymptotic power properties are given in [1]. As in Rothman [9] the difficulty in modifying these tests in the toroidal case is that there is no natural origin for the distribution functions on a circle. Moreover, different arbitrary starting points give the test statistics $A_n$ and $B_n$ different values. In this paper we propose that the statistic $C_n$ be used for our problem, with the surface of the torus replacing the plane. \begin{equation*}\tag{8}C_n = n \int\int Z_n^2(x, y) dF_n(x, y)\end{equation*} where $Z_n(x, y) = T_n(x, y) + \int\int T_n(x, y) dF_n(x, y) - \int T_n(x, y) dF_{n1}(x) - \int T_n(x, y) dF_{n2}(y).$ We note that $C_n$ may be rewritten in the following form which we shall refer to in Section 3: \begin{equation*}\tag{9}C_n = 1/n^2 \sum^n_{j=1} \lbrack\sum^n_{a=1} \{T_n(X_j, Y_j) + T_n(X_a, Y_a) - T_n(X_a, Y_j) - T_n(X_j, Y_a)\}\rbrack^2.\end{equation*} We shall also have occasion to use the random variable, $D_n = n \int\int Z_n^{\ast^2}(x, y) dF_1(x)dF_2(y),$ where $Z_n^\ast(x, y) = T_n(x, y) + \int\int T_n(x, y)dF_1(x)dF_2(y) - \int T_n(x, y)dF_1(x) - \int T_n(x, y) dF_2(y).$ An outline of the paper follows: In Section 2 it is shown that when $H_0$ is true, $C_n - D_n \rightarrow 0$ in probability. The invariance of $C_n$ under changes of origin is proved in Section 3. Finally the asymptotic distribution of $C_n$ under the null hypothesis is obtained in Section 4.

We define the incomplete gamma function ratio for positive $N$ by \begin{align*}\tag{1}P(N, b, X) &= \int^X_0 t^{N-1}e^{-bt} dt/ \int^\infty_0 t^{N-1}e^{-bt} dt \\ &\equiv \int^X_0 D(N, b, t) dt, \text{say},\\ \end{align*} where $N$ and $b$ are positive real numbers and $0 < X \leqq \infty$. In what follows, we generally use the notation $P_N$ to denote the same function, unless $b$ or $X$ are given some specific values. It should be noted that the notation $P(N, b, X)$ does not mean that the function is one of three variables, since we have $P(N, b, X) = P(N, 1, bX)$. The positive real number $b$ is only a scale factor. The function $P_N$ is of appreciable importance in probability and statistics and it is known as the gamma distribution and for $b = \frac{1}{2}$ as the chi-square distribution. It is also of importance in many other branches of applied mathematics. Like the incomplete beta distribution, it is also a special case of the more general confluent hypergeometric function. During the last decade, interest has been revived in the development, use and application of these functions and a number of publications, such as Erdelyi et al. (1953), Tricomi (1954) and Slater (1960), appeared. In these publications, the function treated is defined as the numerator in the middle part of (1) with $b = 1$, and the definition is generally extended for all complex values of $N$ such that $R(N)$ is not a non-positive integer. Many of the properties of $P_N$ are derived in these publications. Bancroft (1949) derived some new properties of the incomplete beta function (particularly, recurrence relations) from the more general properties of the parent hypergeometric function. Similar methods may also be used to derive new properties of $P_N$ of use in statistical work. In this paper, however, we derive such properties directly from the well-known difference-differential properties of $P_N$ given by \begin{align*}\tag{2}(\partial^r/\partial X^r) P_N &= (-b)^r\Delta^rP_{N-r}, \\ &= (\partial^{r-1}/\partial X^{r-1})D(N, b, X),\quad N > r,\\ \end{align*} where $r = 0, 1, 2, 3, \cdots$ and where \begin{equation*}\tag{3}\Delta^{r+1} P_T = \Delta^r(\Delta P_T) = \Delta^r(P_{T+1} - P_T)\end{equation*} are the advancing finite differences of $P_N$ with respect to $N$ and with a unit difference interval. Property (2) for $r = 1$ may be used, in fact, to define the function $P_N$ itself, as in Milne-Thomson (1933). This property was also used in 1947 by Khamis in an unpublished Ph.D thesis (1950) to derive, inter alia, a series expansion in terms of chi-square integrals to approximate to statistical distribution functions and also to formulate a computational method for the tabulation of the chi-square distribution itself. This method was later used by Hartley and Pearson (1950) in computing a five decimal table for this distribution. The chi-square series expansion was later generalized by Khamis (1960a) into an expansion in terms of incomplete gamma function ratios and the computational method was employed by Khamis, (1964a) and (1965) to prepare a six decimal table of the chi-square integral and a master ten decimal table, for very fine $N$ and $X$ intervals, of the incomplete gamma function (1) for $b = \frac{1}{2}$ (a description of this table is given by Khamis (1964b). In this paper we make further use of the property (2) to derive other new and useful properties of the function (1). In Section 2 we derive a simple recurrence relation for $P(T, b, X)$ for $T = N, N + 1$ and $N + 2$. In Section 3 we derive new $N$-wise sum formulae as consequences of this recurrence relation. We then give in Section 4 an $X$-wise sum formula based on the Taylor expansion of (2) in the neighbourhood of $X$. This is made possible by extending the definition of $P(N, b, X)$ for $N \leqq 0$, and thus enabling the removal of the condition $N > r$ in (2) above. Other examples of the use of property (2) are given in Section 5 where in particular, the Laguerre polynomials are expressed in terms of the differences of the function $P(N, b, X)$. Section 5 contains also examples of some numerical applications of the previous results. The importance of the properties derived here and the simplicity inherent in such derivations due to the nature of property (2) are further enhanced by work recently carried out by Wise (1950) and developed further by H. O. Hartley and E. J. Hughes (in process of publication) where the incomplete gamma function ratio is shown to provide quite satisfactory approximations to the incomplete beta function ratio and to many other statistical functions. We note finally, that the notation used in (1) is only one of many notations used by different authors. The references given above include most of the different notations used for the function $P_N$ and other related functions.

Let $X_{ij} (j = 1, 2,\cdots, n_i; i = 1, 2,\cdots, k)$ be $k$ independent samples of sizes $n_1, n_2,\cdots, n_k$ respectively from $k$ populations with continuous cumulative distribution functions $F_1, F_2,\cdots, F_k$ respectively. We assume that the $F_i$'s belong to a family $\mathscr{F}$ of distribution functions indexed by a parameter $\theta$. Let all the $N = \sum^k_{i=1} n_i$ observations be put together and ordered to form a single sequence and suppose that only the first $r$ ordered observations are available. That is, let us have a combined (right) censored sample of total size $r$. Such a censored sample occurs naturally in many physical situations as for example, in problems of life testing where we are interested in comparing the mean life of several physical systems, or in clinical trials or bio-assay problems where we want to compare the efficacy of several drugs but we cannot afford to wait indefinitely to get information on all the sampling units put on test. For details see Basu [2]. For facility of discussion, we shall use the terminology of life testing. Any test based on the first $r$ ordered observations (out of a combined sample of size $N$) will be termed an $r$ out of $N$ test. In this paper we propose two $k$-sample $r$ out of $N$ rank tests which generalize the rank tests proposed by Kruskal [7], Jonckhere [6] and Terpstra [10], [11]. In the first part of the paper we propose the statistic $B_r^{(N)}$ (large values being critical) to test the null hypothesis \begin{equation*}\tag{1.1}H_0:F_1(x) = F_2(x) = \cdots = F_k(x)\end{equation*} (or equivalently, $H_0:\theta_1 = \theta_2 = \cdots = \theta_k = 0$ say, under location alternatives) against the alternative hypothesis \begin{equation*}\tag{1.2}H_1:F_i(x) = F(x, \theta_i) \quad (i = 1, 2, \cdots, k)\end{equation*} In Section 2 we define the statistic $B_r^{(N)}$ and show its relationship with other statistics. The mean and variance of $B_r^{(N)}$ under the null hypothesis is derived in Section 3. In Section 4 we find the asymptotic distribution of $B_r^{(N)}$ both under the null and the non-null case. The computation of $B_r^{(N)}$ has been illustrated by an example in Section 5. In the second part of the paper we consider the statistic $V(N, r)$ (to be defined later) for testing the hypothesis (1.1) against the ordered alternative \begin{equation*}\tag{1.3}H_2:F_1(x) < F_2(x) < \cdots < F_k(x)\end{equation*} for all $x$, where the $F_i(x)$ are labeled in such a way that (1.3) is the ordered alternative being considered. The second formulation is useful in life testing where the $k$ cdf's might be associated with $k$ different processes and the experimenter wishes to test whether the $k$ processes give rise to units with the same life-time distributions against the alternative that the processes can be ordered in a particular manner, in the sense that the $k$ life-time distributions can be ordered uniformly with respect to $x$, (that is, according to their reliabilities). In Section 6 we define the statistic $V(N, r)$ and in Section 7 we have derived the mean and variance of $V(N, r)$ under $H_0$. Section 8 is devoted to investigate the extreme values of $V(N, r)$ and the asymptotic normality of $V(N, r)$ is proved in Section 9. Both the statistics $B_r^{(N)}$ and $V)N, r)$ may be considered as $k$-sample extensions of the $V_r$ statistic proposed by Sobel [9]. The definitions of $B_r^{(N)}$ and $V(N, r)$ are slightly different from what is given earlier in Basu [2].

Associated with certain of the learning models introduced by Bush and Mosteller [1] are random walks $p_1, p_2, p_3, \cdots$ on the closed unit interval with transition probabilities of the form \begin{equation*}\tag{1}P\lbrack p_{n + 1} = p_n + \theta_1(1 - p_n) \mid p_n\rbrack = \varphi(p_n)\end{equation*} and \begin{equation*}\tag{2}P\lbrack p_{n + 1} = p_n - \theta_2p_n \mid p_n\rbrack = 1 - \varphi(p_n)\end{equation*} where $0 < \theta_1, \theta_2 < 1$ and $\varphi$ is a mapping of the closed unit interval into itself. In the experiments to which these models are applied, response alternatives $A_1$ and $A_2$ are available to a subject on each of a sequence of trials, and $p_n$ is the probability that the subject will make response $A_1$ on trial $n$. Depending on which response is actually made, one of two events $E_1$ or $E_2$ ensues. These events are associated, respectively, with the increment $p_n \rightarrow p_n + \theta_1(1 - p_n)$ and the decrement $p_n \rightarrow p_n - \theta_2p_n$ in $A_1$ response probability. The conditional probabilities $\pi_{ij}$ of event $E_j$ given response $A_i$ do not depend on the trial number $n$. Thus (1) and (2) are obtained with $\varphi(p) = \pi_{11}p + \pi_{21}(1 - p)$. Since the linearity of the functions $\varphi$ which arise in this way is of no consequence for the work presented in this paper, we will assume instead simply that \begin{equation*}\tag{3}\varphi \varepsilon C^2(\lbrack 0, 1\rbrack).\end{equation*} We impose one further restriction on $\varphi$ which excludes some cases of interest in learning theory: \begin{equation*}\tag{4}\epsilon_1 = \min_{0 \leqq p \leqq 1} \varphi(p) > 0 \text{and} \epsilon_2 = \max_{0 \leqq p \leqq 1} \varphi(p) < 1.\end{equation*} It follows from a theorem of Karlin ([5], Theorem 37) that under (1)-(4) the distribution function $F^{(n)}_{\theta_1,\theta_2,\varphi}$ of $p_n$ (which depends, of course, on the distribution $F$ of $p_1$) converges as $n$ approaches infinity to a distribution $F_{\theta_1,\theta_2,\varphi}$ which does not depend on $F$. It is with the distributions $F_{\theta_1,\theta_2,\varphi}$ that the present paper is concerned. Very little is known about distributions of this family, though some results may be found in Karlin [5], Bush and Mosteller [1], Kemeny and Snell [6], Estes and Suppes [3], and McGregor and Hui [8]. The only theorem in the literature directly relevant to the present work is one of McGregor and Zidek [9] as a consequence of which, in the case $\theta_1 = \theta_2 = \theta, \varphi(p) \equiv \frac{1}{2}$, $\lim_{\theta \rightarrow 0} \lim_{n \rightarrow \infty} P\lbrack\theta^{-\frac{1}{2}}(p_n - \frac{1}{2}) \leqq x\rbrack = \Phi(8^{\frac{1}{2}}x)$ where $\Phi$ denotes the standard normal distribution function; that is, the distribution $F_{\theta,\theta,\frac{1}{2}}(\theta^{\frac{1}{2}}x + \frac{1}{2})$ converges to a normal distribution as the "learning rate" parameter $\theta$ tends to 0. We will prove, by means of another method, that this phenomenon is of much greater generality. Theorem 1 below shows that, for any positive constant $\zeta$ and any $\varphi$ with $\max_{0 \leqq p \leqq 1}\varphi'(p) < \min (1, \zeta)/\max (1, \zeta)$ there is a constant $\rho$ such that $F_{\theta,\zeta\theta,\varphi}(\theta^{\frac{1}{2}}x + \rho)$ converges to a normal distribution as $\theta \rightarrow 0$. A nonnormal limit is obtained if $\theta_1$ approaches 0 while $\theta_2$ remains fixed as is shown in Theorem 2. In this case $F_{\theta_1,\theta_2,\varphi}(\theta_1x)$ converges to an infinite convolution of geometric distributions. If $f(p,\theta) = p + \theta(1 - p)$ then (1) and (2) can be written in the form \begin{equation*}\tag{5}P\lbrack p_{n + 1} = f(p_n, \theta_1) \mid p_n\rbrack = \varphi(p_n)\end{equation*} and \begin{equation*}\tag{6}P\lbrack p_{n + 1} = 1 - f(1 - p_n, \theta_2) \mid p_n\rbrack = 1 - \varphi(p_n).\end{equation*} In Section 4 it is shown that the linearity of $f(p,\theta)$ in $p$ and $\theta$ is not essential to the phenomena discussed above. Theorems 3 and 4 present generalizations of Theorems 2 and 1, respectively, to "learning functions" $f(p,\theta)$ subject only to certain fairly weak axioms. A somewhat different learning model, Estes [2] $N$-element pattern model, leads to a finite Markov chain $p_1, p_2, p_3, \cdots$ with state space $S_N = \{jN^{-1}: j = 0, 1, \cdots, N\}$ and transition probabilities \begin{align*}\tag{7}P\lbrack p_{n + 1} &= p_n + N^{-1} \mid p_n\rbrack = \varphi(p_n), \\ \tag{8}P\lbrack p_{n + 1} &= p_n - N^{-1} \mid p_n\rbrack = \psi(p_n),\\ \end{align*} and \begin{equation*}\tag{9}P\lbrack p_{n + 1} = p_n \mid p_n\rbrack = 1 - \varphi(p_n) - \psi(p_n)\end{equation*} where $\varphi(p) = c\pi_{21}(1 - p), \psi(p) = c\pi_{12}p, 0 < c \leqq 1$, and for the sake of this discussion we suppose that $0 < \pi_{12}, \pi_{21} < 1$. In this case a limiting distribution $F_{N, \varphi,\psi}$ of $p_n$ as $n \rightarrow \infty$ exists and is independent of the distribution of $p_1$ by a standard theorem on Markov chains. Estes [2] showed that the limit is binomial over $S_N$ with mean $r = \pi_{21}/(\pi_{12} + \pi_{21})$. It then follows from the central limit theorem that $\lim_{N \rightarrow \infty}\lim_{n \rightarrow \infty} P\lbrack N^{\frac{1}{2}}(p_n - r) \leqq x\rbrack = \Phi\lbrack x/(r(1 - r))^{\frac{1}{2}}\rbrack.$ In Section 5 it is shown that our method permits an extension of this result to much more general $\varphi$ and $\psi$.

This paper deals with linear regressions \begin{equation*}\tag{1.1}y_k = x_{k1}\beta_1 + \cdots + x_{kq}\beta_q + \epsilon_k, \quad k = 1, 2, \cdots\end{equation*} with given constants $x_{km}$ and with error random variables $\epsilon_k$ that are (a) uncorrelated or (b) independent. Let $E\epsilon_k = 0, 0 < E\epsilon^2_k < \infty$ for all $k$. The individual error distribution functions (d.f.'s) are not assumed to be known, nor need they be identical for all $k$. They are assumed, however, to be elements of a certain set $F$ of d.f.'s. Consider the family of regressions associated with the family of all the error sequences possible under these restrictions. Then conditions on the set $F$ and on the $x_{km}$ are obtained such that the least squares estimators (LSE) of the parameters $\beta_1, \cdots, \beta_q$ are consistent in Case (a) (Theorem 1) or asymptotically normal in Case (b) (Theorem 2) for every regression of the respective families. The motivation for these theorems lies in the fact that under the given assumptions statements based only on the available knowledge must always concern the regression family as a whole. It will be noticed moreover that the conditions of the theorems do not require any knowledge about the particular error sequence occurring in (1.1). Most of the conditions are necessary as well as sufficient, with the consequence that they cannot be improved upon under the limited information assumed to be available about the model. Since the conditions are very mild, the results apply to a large number of actual estimation problems. We denote by $\mathfrak{F}(F)$ the set of all sequences $\{\epsilon_k\}$ that occur in the regressions of a family as characterized above. Thus, $\mathfrak{F}(F)$ comprises all sequences of uncorrelated (Case (a)) or independent (Case (b)) random variables whose d.f.'s belong to $F$ but are not necessarily the same from term to term of the sequence. For each $G \varepsilon F$ the relations $\int x dG = 0$ and $0 < \int x^2 dG < \infty$ hold. In this paper, $\mathfrak{F}(F)$ may be looked upon as a parameter space. A parameter point then is a sequence of $\mathfrak{F}(F)$. Correspondingly, we say that a statement holds on $\mathfrak{F}(F)$ (briefly on $F$) if it holds for all $\{\epsilon_k\} \varepsilon \mathfrak{F}(F)$. The statements of Theorems 1 and 2 are of this kind. The proof of Theorem 1, as well as the proof of the sufficiency in Theorem 2, is elementary and straight forward. Theorem 2 is a special case of a central limit theorem (holding uniformly on $\mathfrak{F}(F)$) for families of random sequences [3]. Some similarity between the roles of the parameter spaces $\mathfrak{F}(F)$ in our theorems and of the parameter spaces that occur, e.g., in the Gauss-Markov and related theorems may be seen in the fact that these theorems remain true only as long as the conclusions in the theorems hold for every parameter point in the respective spaces. As is well known, the statements in the Gauss-Markov and related theorems hold for every parameter vector $\beta_1, \cdots, \beta_q$ in a $q$-dimensional vector space (see e.g. Scheffe 1959, p. 13, 14). A result in the theory of linear regressions that bears some resemblance with the theorems of this paper has been obtained by Grenander and Rosenblatt (1957, p. 244). Let the error sequence $\{\epsilon_k\}$ in (1.1) be a weakly stationary random sequence with piecewise continuous spectral density, and let the regression vectors admit a joint spectral representation. Under these assumptions Grenander and Rosenblatt give necessary and sufficient conditions for the regression spectrum and for the family of admissible spectral densities in order that the LSE are asymptotically efficient for every density of the family. In Sections 3 and 6 we discuss some examples relevant to Theorems 1 and 2.

This study was initiated in connection with estimating parameters involved in a certain stochastic process of population growth. Because of the nature of distribution functions arising in such studies, the usual methods of estimation result in formulas which are so complex that it is difficult, if not impossible, to obtain explicit solutions for the estimates of the parameters. Investigation of the problem led to an extension of the method of best asymptotically normal estimates developed by Neyman [1]. The estimates derived are termed regular best asymptotically normal estimates (RBAN estimates). This extension can be applied to other problems. In [1], Neyman considers a whole class of estimates which possess the properties of consistency, of asymptotic normality, and of asymptotic efficiency, and he provides estimates having these asymptotic properties for the case of multinomial distributions. His method is extended in the present paper to a more general case in which random vectors are dealt with. Such an extension was considered by Barankin and Gurland [2], who studied a large class of estimates and showed that if the distributions involved are members of Koopman's family, it is still possible to reach the Cramer-Rao lower bound. The purposes of the present paper are to discuss a subclass of the estimates considered by Barankin and Gurland and to present simple methods of generating such estimates. The estimates discussed are based on a number of independent random vectors whose distribution functions are not specified. It is proved that under certain regularity conditions, the regular and consistent estimates obtained are asymptotically normal as the number of random vectors tends to infinity. A necessary and sufficient condition for a regular and consistent estimate to have a "minimal" asymptotic covariance matrix is given. An expression is derived for the "minimal" asymptotic covariance matrix. It is also proved that if a function $\mathbf{f}$ satisfies certain conditions, then in order that $\mathbf{f}(\tilde\theta)$ be an RBAN estimate of $\mathbf{f}(\theta)$ at $\mathbf{f}(\theta^0)$, where $\theta^0$ is the true value of the parameter point $\theta$, it is necessary and sufficient that the argument $\tilde\theta$ be an RBAN estimate of $\theta$ at $\theta^0$. Methods of generating RBAN estimates are given. For simplicity of presentation, matrix notation is used throughout this paper. By derivatives of a matrix with respect to a vector (or with respect to a second matrix) is meant the derivatives of the matrix simultaneously with respect to all the components of the vector (or all the elements of the second matrix). The usual rules of differentiation with respect to vectors are used.

Let $(x_{nk}), k = 1, 2, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e. $\lim_{n \rightarrow \infty} \max_{1 \leqq k \leqq k_n} P(|x_{nk}| > \epsilon) = 0$ for every $\epsilon > 0$, and for each $n, x_{n1}, \cdots, x_{nk_n}$ are independent). Let $S_n = x_{n1} + \cdots + x_{nk_n} - A_n$ where the $A_n$ are constants and let $F_n(x)$ be the distribution function of $S_n$. In a previous paper [3] the system of infinitesimal, rowwise independent random variables $(|x_{nk}|^r)$ was studied for $r \geqq 1$. Specifically, let $S^r_n = |x_{n1}|^r + \cdots + |x_{nk_n}|^r - B_n(r),$ where the $B_n(r)$ are suitably chosen constants. Let $F_n^r(x)$ be the distribution function of $S^r_n$. Necessary and sufficient conditions for $F_n^r(x)$ to converge $(n \rightarrow \infty)$ to a distribution function $F^r(x)$ and for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ were given, together with the form that $H(x)$ must take. In Section 2 of this paper we consider the system $(|x_{nk}|^r)$ for $0 < r < 1$. Results similar to the above are found, replacing $(r \rightarrow \infty)$ by $(r \rightarrow 0^+)$. However different assumptions must be made at certain points. Various remarks are made in this paper to show where the results here differ from [3]. In particular it is shown that, if $F^r(x)$ converges $(r \rightarrow 0^+)$ to a distribution function $H(x)$, then $H(x)$ will be the distribution function of the sum of two independent random variables, one Poisson and the other Gaussian. Furthermore, while the Gaussian summand may or may not be degenerate, the Poisson summand will be nondegenerate in all but one special case.

Let $(X^{(1)}_{j1}, \cdots, X^{(p)}_{j1}), \cdots, (X^{(1)}_{jn_j}, \cdots, X^{(p)}_{jn_j})$ be random samples of size $n_j$ from the $c$ continuous $p$-variate distribution functions $F_j(\mathbf{x}) = F(\mathbf{x} - \mathbf{\theta}_j)$ where $\mathbf{x} = (x_1, \cdots, x_p), \mathbf{\theta}_j = (\theta_{ij}, \cdots, \theta_{pj})$ and $j = 1, \cdots, c$. This paper is concerned with the problem of testing the hypothesis $H : \mathbf{\theta}_1 = \mathbf{\theta}_2 = \cdots = \mathbf{\theta}_c$ against the alternative that all $\mathbf{\theta}_j$ are not equal on the basis of the above $c$ samples. When we especially consider the asymptotic efficiency of test, the following alternative $K$ is adopted, $K : \mathbf{\theta}_j = \mathbf{\theta} + \mathbf{\nu}_j/N^{\frac{1}{2}},\quad\mathbf{\nu}_j = (\nu_{ij}, \cdots, \nu_{pj}),\quad j = 1, \cdots, c.$ We shall develop in this paper some test procedures for the hypothesis $H$ which are originated from the paper of Chernoff-Savage [2]. When $c = 2$, that is multivariate two-sample tests, Sugiura [7] has proposed a test statistic of Wilcoxon type. On the other hand, Puri [5] has derived univariate several-sample tests in general type including the test of Kruskal-Wallis [4]. It will be shown that the latter are corresponding to the case $p = 1$ in this paper and the former is a particular one among the case $c = 2$ of our tests.

In two recent papers, Paulson [1] and Mosteller [2] have called attention to several unsolved problems in $k$-sample theory. A problem which is typical of the ones considered in this paper is as follows. Let $\pi_1, \pi_2, \cdots, \pi_k$ be a set of normal populations, $\pi_i$ having an unknown mean $m_i$ and variance $\sigma^2, G(x, \theta_i)$ being the distribution function which characterizes $\pi_i$. Samples of equal size are drawn from each population, $\bar X_i$ being the sample means, and $S^2$ the estimate of $\sigma^2$ obtained. The problem is to construct a suitable decision rule $d = d(\{\bar X_i\}; S^2)$ to select one or more populations, the object being to minimize the expected value of the random distribution function $G(x \mid s(d)) = \sum^k_{i=1} Z_i(d) \cdot G(x, \theta_i) / \sum^k_{i=1} Z_i(d),$ where $Z_i(d) = 1$ if $\pi_i$ is selected by $d$, and $= 0$ otherwise. It is shown that under the restriction of impartial decision, the rule $d_k =$ "Always select only the population corresponding to the greatest $\bar X_i$" cannot be improved, no matter what $x$ or the true parameter values may be. It follows (i) that $d_k$ is the uniformly best decision rule in the class of impartial decision rules for all weight functions of type $W = \max_i \{m_i\} - \big(\sum^k_{i=1} z_im_i / \sum^k_{i=1} z_i\big)$, and (ii) that the customary $F$ and $t$ tests of analysis of variance are not relevant to the problem. This result is an application of Theorem 1 which applies to a number of similar problems concerning $k$ populations, especially when the populations admit sufficient statistics for their parameters. Two examples of statistical applications are given in Section 6.

The Bayesian approach to statistical problems, though fruitful in many ways, has been rather unsuccessful in treating nonparametric problems. This is due primarily to the difficulty in finding workable prior distributions on the parameter space, which in nonparametric ploblems is taken to be a set of probability distributions on a given sample space. There are two desirable properties of a prior distribution for nonparametric problems. (I) The support of the prior distribution should be large--with respect to some suitable topology on the space of probability distributions on the sample space. (II) Posterior distributions given a sample of observations from the true probability distribution should be manageable analytically. These properties are antagonistic in the sense that one may be obtained at the expense of the other. This paper presents a class of prior distributions, called Dirichlet process priors, broad in the sense of (I), for which (II) is realized, and for which treatment of many nonparametric statistical problems may be carried out, yielding results that are comparable to the classical theory. In Section 2, we review the properties of the Dirichlet distribution needed for the description of the Dirichlet process given in Section 3. Briefly, this process may be described as follows. Let $\mathscr{X}$ be a space and $\mathscr{A}$ a $\sigma$-field of subsets, and let $\alpha$ be a finite non-null measure on $(\mathscr{X}, \mathscr{A})$. Then a stochastic process $P$ indexed by elements $A$ of $\mathscr{A}$, is said to be a Dirichlet process on $(\mathscr{X}, \mathscr{A})$ with parameter $\alpha$ if for any measurable partition $(A_1, \cdots, A_k)$ of $\mathscr{X}$, the random vector $(P(A_1), \cdots, P(A_k))$ has a Dirichlet distribution with parameter $(\alpha(A_1), \cdots, \alpha(A_k)). P$ may be considered a random probability measure on $(\mathscr{X}, \mathscr{A})$, The main theorem states that if $P$ is a Dirichlet process on $(\mathscr{X}, \mathscr{A})$ with parameter $\alpha$, and if $X_1, \cdots, X_n$ is a sample from $P$, then the posterior distribution of $P$ given $X_1, \cdots, X_n$ is also a Dirichlet process on $(\mathscr{X}, \mathscr{A})$ with a parameter $\alpha + \sum^n_1 \delta_{x_i}$, where $\delta_x$ denotes the measure giving mass one to the point $x$. In Section 4, an alternative definition of the Dirichlet process is given. This definition exhibits a version of the Dirichlet process that gives probability one to the set of discrete probability measures on $(\mathscr{X}, \mathscr{A})$. This is in contrast to Dubins and Freedman [2], whose methods for choosing a distribution function on the interval [0, 1] lead with probability one to singular continuous distributions. Methods of choosing a distribution function on [0, 1] that with probability one is absolutely continuous have been described by Kraft [7]. The general method of choosing a distribution function on [0, 1], described in Section 2 of Kraft and van Eeden [10], can of course be used to define the Dirichlet process on [0, 1]. Special mention must be made of the papers of Freedman and Fabius. Freedman [5] defines a notion of tailfree for a distribution on the set of all probability measures on a countable space $\mathscr{X}$. For a tailfree prior, posterior distribution given a sample from the true probability measure may be fairly easily computed. Fabius [3] extends the notion of tailfree to the case where $\mathscr{X}$ is the unit interval [0, 1], but it is clear his extension may be made to cover quite general $\mathscr{X}$. With such an extension, the Dirichlet process would be a special case of a tailfree distribution for which the posterior distribution has a particularly simple form. There are disadvantages to the fact that $P$ chosen by a Dirichlet process is discrete with probability one. These appear mainly because in sampling from a $P$ chosen by a Dirichlet process, we expect eventually to see one observation exactly equal to another. For example, consider the goodness-of-fit problem of testing the hypothesis $H_0$ that a distribution on the interval [0, 1] is uniform. If on the alternative hypothesis we place a Dirichlet process prior with parameter $\alpha$ itself a uniform measure on [0, 1], and if we are given a sample of size $n \geqq 2$, the only nontrivial nonrandomized Bayes rule is to reject $H_0$ if and only if two or more of the observations are exactly equal. This is really a test of the hypothesis that a distribution is continuous against the hypothesis that it is discrete. Thus, there is still a need for a prior that chooses a continuous distribution with probability one and yet satisfies properties (I) and (II). Some applications in which the possible doubling up of the values of the observations plays no essential role are presented in Section 5. These include the estimation of a distribution function, of a mean, of quantiles, of a variance and of a covariance. A two-sample problem is considered in which the Mann-Whitney statistic, equivalent to the rank-sum statistic, appears naturally. A decision theoretic upper tolerance limit for a quantile is also treated. Finally, a hypothesis testing problem concerning a quantile is shown to yield the sign test. In each of these problems, useful ways of combining prior information with the statistical observations appear. Other applications exist. In his Ph. D. dissertation [1], Charles Antoniak finds a need to consider mixtures of Dirichlet processes. He treats several problems, including the estimation of a mixing distribution, bio-assay, empirical Bayes problems, and discrimination problems.

In a single-server first-come-first-served queue the waiting-times of successive customers are related by the equation \begin{equation*}\tag{1}w_{n + 1} = \lbrack w_n + U_n\rbrack^+\end{equation*} where \begin{equation*}\tag{2}U_n = S_n - T_n.\end{equation*} Here $S_n$ and $T_n$ are the service-time of the $n$th customer, and the time between the arrivals of the $n$th and the $(n + 1)$th customers, respectively. In the particular case of mutually independent identically distributed $U_n$ the basic investigation of this equation was carried out by Lindley [1], who found a simple necessary and sufficient condition for the existence of a stationary waiting-time distribution and derived this distribution in certain special cases. The theory was developed by Smith [2], who under fairly weak conditions gave a systematic treatment of the Wiener-Hopf equation obtained by Lindley. In the less restricted case when $U_n$ is a strictly stationary process it has been shown by the author (Loynes [3]) that under a simple condition the existence of a unique stationary distribution is again ensured: it is the purpose of this paper to show how (when this condition is satisfied) the stationary distribution may sometimes be found, and to obtain some qualitative results. The theory will be developed in Sections 3 and 4, and some examples discussed in Section 5. On the assumption that $U_n$ is a metrically transitive sequence the condition just referred to is \begin{equation*}\tag{3}E\lbrack U_n\rbrack < 0,\end{equation*} and we shall therefore suppose this satisfied throughout the paper. The assumption that $U_n$ is metrically transitive does not affect most of the arguments in this paper, but it will be made for convenience. For the existence of a unique stationary distribution (3) is both necessary and sufficient, but there is sometimes more than one stationary distribution when the inequality in (3) is replaced by equality. Such situations have been ignored here for several reasons: they are not very common, some of the arguments either break down or need adaptation, and in any case the problems are not then as difficult, for it follows from equation (7) of [3] that the function $\psi$ occurring in (5) below is then identically equal to one, so that the only unknown is $\phi$. There is never a stationary distribution when the inequality in (3) is reversed. Throughout the entire investigation we shall suppose the queue has a structure which satisfies the following condition (H). This is actually no restriction at all when the queue is in the stationary state, since the variables $z_n$ occurring in it may be taken as $(W_{n + 1}, T_{n + 1})$, where $\{W_n\}$ is the stationary waiting-time sequence. However, all the results will refer to the distribution of the waiting-time conditional on $z_n$, so that it will always be tacitly assumed that the distribution of $z_n$ is known in order to allow deductions about the unconditional distribution of the waiting-time. (H): There exists a sequence $\{z_n\}$ (defined on the same probability space as $w_n$) of random vectors--i.e., in finite-dimensional Euclidean space--with the following properties: (i) $\{z_n, S_n, T_n\}$ is a strictly stationary process. (ii) $S_n, T_n$, and $w_n$ are conditionally independent given $z_{n - 1}, z_n$. (iii) $w_n$ and $z_n$ are conditionally independent given $z_{n - 1}$. The assumption that (H) is satisfied allows us to write down the equation satisfied by the stationary (conditional) waiting-time distribution function. We cannot, however, in such a general situation actually find the solution to the equation, and to discover appropriate further restrictions which may make this possible we may consider further the case treated by Lindley and Smith. The general solution given by Smith was analytically complex and apparently only obtainable from the Wiener-Hopf technique or the equally specialized method due to Spitzer [4]. If, however, either the service-times or the inter-arrival times had a distribution with a rational characteristic function the solutions were much simpler in form, and in at least a few cases had been obtained previously by other methods. With this in mind we impose similar conditions here on the conditional characteristic functions; on that of the interarrival time in Section 3, and on that of the service-time in Section 4. The lack of success in treating the problem without these restrictions is probably no great loss in practice, in view of the complex nature of the Wiener-Hopf solution for the simple case. The results obtained under the conditions imposed so far are in the nature of aids to the solution of the problem, rather than the solution itself. A simple situation for which the required distribution may then be obtained straightforwardly occurs when the following "finite-matrix" condition is satisfied: (HF): Condition (H) holds, and $z_n$ has (with probability one) only a finite number $k$ of different possible values. An interesting result obtained by Smith showed that, roughly speaking, the characteristic functions of the waiting-time and the service-time had the same number of poles, and a somewhat similar result related the distribution of the inter-arrival times to that of the waiting-time. Under (HF) it will be found that similar results are also true. Smith in fact gave explicit formulae of a simple type for the two special cases, but we shall content ourselves here with pointing out how corresponding formulae could be obtained, since it appears that they are no longer simple when $k$ has a value other than one. As we have already remarked, all simple queues are included in (H) when in the stationary state. Even under the condition that the distribution of $z_n$ be known, a large class of queues satisfies (H), such as that in which $S_n$ and $T_n$ are independent stationary Markov processes. An example of a queue of considerable interest which apparently does not admit a specification of this type is that in which customers are due at regular intervals but are late independently with a lateness distribution which extends to $\infty$ (if the lateness distribution and service time distribution are both negative exponential, then (H) is satisfied, and a brief discussion of this example will be found in Section 5). It will be seen that by taking $z_n = 1$ the classical case with independence is included under (HF); two other types of queue satisfying (HF) are of some interest, and are discussed in Section 5--one being a simple queue whose input is a mixture of two streams of customers, and the other a queue whose input consists of customers who have already been served in a first queue. In addition to any other queues which may fall into this class, it is presumably true, that any queue may be approximated by those of this "soluble" class, although there are of course very great difficulties in deciding how such an approximation should be carried out. Finally we observe that since the content of a semi-infinite dam in discrete time is described by the same equation (1), the results obtained here may be applied in that context also, and in particular the finite-matrix case gives a class of dams with serially dependent inputs for which explicit solutions can be found.

In order to extend Maxwell's later method of developing the dynamical theory of a gas to cases other than that which he considered (viz., a gas whose molecules are point centres of repulsive force varying inversely as the fifth power of the distance), a knowledge of the velocity distribution function, in the disturbed state of the gas, is necessary. In this paper the simplest possible form is assumed for the function, consistent with the fulfilment of certain preliminary conditions. This form is ( hm / π ) 3/2 e - hm ∑( u - u 0 ) 2 {1 + F ( u - u 0 , v - v 0 , w - w 0 )}, where F is a polynomial, in the three variables indicated, of the third degree. The theory of viscosity and thermal conduction, in simple and mixed gases, is developed without assuming any property of the molecules beyond that of spherical symmetry. Perhaps the most interesting result is the relation between the viscosity μ , the thermal conductivity ϑ, and the specific heat at constant volume, C v , for a simple monatomic gas, viz., ϑ = 5/2 μ C v .

0 1. SUMMARY This paper deals with the distribution function of the order statistics xn_m+l(m = 1, 2 ... ), Mn,m(x). For this distribution function of the mth largest value (if m is counted from above) approximate formulae are derived. These formulae are generalizations of the corresponding approximate formulae for the distribution of the extremes proper. Successively the initial distributions f(x) are supposed to be of exponential type, Cauchy type or of limited type (finite range). We first deal with the basic conditions to be imposed on the initial distributionsf(x). An expansion formula has been derived for the distribution of the excess, 1- F(x), which plays an important part in the investigations of this paper. We then consider the general formula of the initial distribution of the mth values, Mn,m(x). Appropriate formulae have been derived to determine the mode and the maximum value of Mn,m(x). The behaviour of the maximum value by varying n and m has also been studied and approximate formulae for Mn, m(x) are successively deduced. Every succeeding formula has a more restricted range of application. Finally, limiting expressions for Mn m(X) are deduced for the three types of initial distributions, mentioned above. The well-known limiting functions of Fisher-Tippett and Gumbel are deduced again. Formulae for the determination of the interval of application have been deduced.

The present author suggests a new distribution function for the roots of the complex polynomial corresponding to the nulls of the radiation pattern of a linear antenna array on the unit circle in a complex plane which is more general than the one suggested by the same author in a previous paper. The new distribution function is of the form ψ K =ψ 0 {1+ξK ((ln(ψ m /ψ 0 -1)-lnξ)/(ln m)) }. When ψ 0 and ξ are chosen properly, the suggested distribution includes the following distributions or arrays as special cases: (1) the uniform end-fire arrays; (2) the uniform broadside arrays; (3) the Schelkunoff's distribution in the range of ψ; (4) the Dolph-Tchebyscheff's distribution for number of elements not greater then seven. But the suggested distribution has the advantage of improving all of them, especially for the suppression of side-lobes near-by the main beam which is very important in certain practical applications.Moreover, since the primary patterns of many types of radiating elements fall off with increasing angle from the main maximum so that the wide-angle lobes would be at a lower level than those close to the main beam, it is necessary to design an array pattern in such a way so that the side-lobes should increase in magnetude with increase in angle from the position of the main beam in order to achieve an over-all pattern with side-lobes all equal or approximately equal in magnitude which is useful in many practical applications. Neither Schelkunoff's distribution in the range of ψ nor Dolph-Tchebyscheff's distribution meets this requirement but the suggested distribution does.In this paper, explicit expressions for the relative antenna currents have been put in closed forms convenient for calculation and semigraphical method has been used for all the pattern calculations.