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SUMMARY This paper contains a study of the departure process from M/G/1 queues with a waiting space of size N> 0. The property of the departure process which receives the most attention is the covariance of pairs of departure intervals, although a more thorough investigation of the probabilistic structure is undertaken for the case N= 1. A main result of the paper is that in the M/G/1 queue with N = 1, departure intervals separated by one or more intervals are independent. When G is the deterministic server, the departure stream is a renewal process. The paper concludes with an expression for the covariance of any pair of intervals in the departure process for any value of N, and numerical values for correlations in the case of a deterministic server with N= 2.

1. SUMMARY In this paper the main discussion is concerned with obtaining asymptotic results for sequences of birth and death processes which are similar to the central limit theorem for sequences of univariate random variables. The motivation is the need to obtain useful approximations to the distributions of sample paths of processes which arise as models for population growth, but for which Kolmogorov differential equations are intractable. In the first section, univariate processes are considered, and conditions are given for the weak convergence of Z N (t) = {X N (t) - aN}/N, where {X N (t), N = 1,2,…} is a sequence of ergodic birth and death processes, to those of an Ornstein-Uhlenbeck process N → ∞. A heuristic method is given which may help explain why this convergence holds, and some examples are given for purposes of illustration. The second part deals with multivariate processes, and three examples are considered in detail: a model for the growth of the sexes in a biological population, a multivariate Ehrenfest process, and a model for the growth and interreaction of two cities. The paper concludes with a discussion of various related results. It is shown that in certain special cases it is possible to obtain diffusions other than the Ornstein-Uhlenbeck process as limits. Finally, heavy traffic results are included for congestion situations originally considered in the special case of time-homogenous arrival rates by Kingman. Transient processes such as epidemics are also shown to exhibit a “central limit” behavior.

Scheffe, in two recent papers (1958, 1963), has given designs for experimenting with mixtures. The basis of these designs is the choice of symmetrical arrangements of points in the factor space and the fitting of carefully chosen models which have exactly as many coefficients as there are data points. The designs are such that some of the experiments do not contain any of one or more ingredients of the mixture. This may or may not be a disadvantage depending on the problem involved. Here an alternate form of design selection is made for the case of mixtures of four factors. It extends the method used in the three factor case in an earlier paper, (Draper and Lawrence, 1965). This involves the premise that, in the absence of specialist knowledge about the form of the true response function, it is desired to fit a response surface equation of first or second order over the factor space of possible mixtures, and experimental runs are needed which, in a certain sense, ensure the best surface fit possible. The principles used in the choice of appropriate designs will be those originally introduced by Box and Draper (1959). (Author)

H. Ruben (1966) has suggested a simple approximate normalization for the correlation coefficient in normal samples, by representing it as the ratio of a linear combination of a standard normal variable and a chi variable to an independent chi variable and then using Fisher's approximation to a chi variable. This result is extended in this paper to a matrix, which in a sense is the correlation coefficient between two vector variables x and y. The result is then used to obtain large sample null and non-null (but in the linear case) distributions of the Hotelling-Lawley criterion and the Pillai criterion in multivariate analysis. Williams (1955) and Bartlett (1951) have derived some exact tests for the goodness of fit of a single hypothetical function to bring out adequately the entire relationship between two vectors x and y, by factorizing Wilks' lambda suitably. These factors are known as 'direction' and 'collinearity' factors, as they refer to the direction and collinearity aspects of the null hypothesis. In this paper, the other two criteria viz. the Hotelling-Lawley and Pillai criteria are partitioned into direction and collinearity parts and large sample tests corresponding to them are derived for testing the goodness of fit of an assigned function.

SUMMARY The papers by Dempster and Williams in this issue are discussed. DR DEMPSTER'S paper is valuable in that it sets up a clear model in terms of which some aspects of the fiducial argument can be discussed. However, the model in question does not accurately represent the situation contemplated by Fisher in this connection. Although Fisher expressed himself several times as averse to certain formalist axiomatic versions of the process of statistical inference, this was not because he had any fundamental objection to formalization, but rather because of his recognition, (a) of the limitations of formal processes and (b) of the fact that any formalization of inductive processes would be exceedingly complex-much more complex than formalizations of mathematical arguments. In any system so far considered by formal logicians (see, for example, Tarski, 1930) the set of consequences 6(S) of a set of S propositions is an increasing function of X, in the sense that if S' includes S, then V(S') includes V(S). Fisher has many times pointed out that such a property is not true in inductive inference. Another point is that in miathematical logic one is normally concerned with the property of "derivability" of a proposition, rather than with the property of "truth". Tarski (1933) showed that a notion of "truth" could be introduced into a formal system, provided this was part of a hierarchy of systems, each with its "truth" notion, applying to systems lower in the hierarchy. Any formal account of probabilistic inference will require a further notion of "knowledge". The proposition "p is known" will clearly have to imply the proposition "p is true", just as the proposition "p is derivable" implies the proposition "p is true" (given that the axioms are true). Again, a sequence of formal systems will need to be contemplated, corresponding to the fact that the class of propositions p which are known changes with time. Anyone familiar with the work of mathematical logicians now concerned with the difficult problems of semantics will feel confident that in time we shall attain to greater formal clarity on these matters, but an awareness of the difficulties already met with in connection with truth and meaning will make anyone cautious in expecting much progress in connection with "knowledge" in the near future. We are therefore left with the less completely formalized situations which need to be supplemented by judgements concerning their applicability in given cases. Such a model, applicable to the fiducial argument, is one in which, as indicated by

SUMMARY We say that a single server queue is in "heavy traffic" when the traffic intensity p is less than, but very near, unity. Then the equilibrium waiting time w will be large, and it is proved that, under certain conditions, the It is therefore of some interest to ask if there are any results which hold under somewhat weaker conditions. It is not to be expected that these will be results valid in general, but rather that they will hold in some limiting case. The limiting case to be considered here is that usually known as "heavy traffic". A common way of formulating queueing problems with a single server is in terms of a traffic intensity p defined as the average number of customers arriving during a service period. For p > 1 the queue will be stable, while for p > 1 its length will increase indefinitely as time goes on. In particular, when p < 1 an equilibrium waiting-time distribution will exist, and as p approaches 1 from below, this waiting time will become long. This situation, when p is less than, but near, unity, we describe as "heavy traffic". The main conclusion of the present paper may roughly be stated as follows: If w is the equilibrium waiting time, then the distribution of (1- p)w is asymptotically negative exponential as p tends to 1. The parameter of this exponential distribution is given in terms of the means and variances of the input and service processes. In the particular case of the queue GI/G/l the result has been proved by the author (1961). We shall first indicate heuristically why this result is to be expected, and what conditions it is necessary to impose. A rigorous proof is then given for a wide class of queues, and this is shown to include a number of cases of interest. The correspond- ing analysis for the queue length and the busy period is also briefly discussed. All the work in this paper may be re-interpreted in the usual way in terms of random walks or semi-infinite dams.

SUMMARY A single-server system with stationary compound Poisson input and general independent service times, the latter being subject to random interruptions of independently but otherwise arbitrarily distributed durations, is studied. For a variety of service-interruption interactions (including the preemptiverepeat) the distributions of busy period duration, of queue length, and of waiting time are characterized by transforms and by moments. Applications are made to priority scheduling problems. MANY situations in which waiting lines develop are characterized by the occurrence of interruptions in customer service. Such interruptions may be caused by breakdowns of a machine that provides service, for example, an electronic computer. Also, if certain customers are assigned priority, then the appearance of one of these may bring about an interruption in the servicing of low-priority customers. In practice it would not be surprising to find systems that experience interruptions of both sorts. In this paper we consider the effect of service interruptions upon a waiting-line process of the following kind: customers appear in accordance with a stationary compound Poisson process (i.e. bunches of customers arrive randomly), and are served in turn by a single facility. The basic customer servicing times are independently, identically, but otherwise arbitrarily, distributed. Interruptions appear at random, in the sense that, if the system is currently free of interruption, the time until the next interruption occurs is exponentially distributed. Interruption durations are identically, independently and arbitrarily distributed. Without interruptions the process described has been discussed by Gaver (1959); the present paper is an adaptation of the approach of the latter paper to the needs of the interruption problem. Previous treatments of similar problems, emphasizing priorities, have been given by Cobham (1954), Stephan (1956), Kesten and Runnenberg (1957), White and Christie (1958), Morse (1958, Chapter 9) and Miller (1960). The influence of service interruptions upon waiting-line behaviour cannot be investigated without specifying in detail the interaction between the interruption process and the service process. Throughout the present paper it will be assumed that all interruptions occurring during a particular customer's service period must take effect during, or immediately after, that period. Thus interruptions may be preemptive, summarily breaking in upon a service in progress, or postponable to the end of that period, but not beyond. If interruptions are preemptive it may be possible to resume service from the point at which interruption took place when the interruption is cleared; such an interaction (between service and interruption) is called

SUMMARY THE solution is given to the problem of determining the efficiency of N machines looked after by an operative who walks round them in one direction, always taking the same time to walk to a given machine from the previous one and to inspect it and service it (we call this overall time the "walking time"), and then, if the machine has stopped, taking a fixed time to repair it. neglecting these points, Ashcroft (1950) gives the efficiency for variable repair times and extensive tables for constant repair times. Benson & Cox (1951) give further details for distributed repair times and groups of machines attended by teams of operatives, and they also discuss the problem of ancillary work, that is work other than the repair of stopped machines. They deal approximately with what they call "spread" ancillary work (which we call "servicing") by an adjustment of one of their parameters. These papers also assume that the operative attends to the machines in the order in which they stopped, and so any systematic patrolling is excluded. In the present paper we discuss a problem more realistic in some work study applications, though to do this we assume a regular patrolling system (somewhat different from that discussed by Brunnschweiler, 1954). The operative walks round the group of machines in a strictly defined order, and repairs and restarts any machines which are found stopped, passing after inspecting and/or servicing any machine which is running. We assume that the operative takes a fixed time to walk from a given machine to the next, (though this may be different for different pairs of machines) and that the repair or clearing times are constant for each stoppage and each machine. Further, we assume that the machines stop at random with the same mean frequency and indepen- dently of each other. The difficulty in dealing with these problems lies in calculating the effect of "interference" which is the term sometimes used for the incidence of simultaneous stoppages at several machines. In fact, if the optimum number of machines is being controlled by one operative, interference will often be a marked feature. No simple or approximate treatment of the problem appears possible if account of interference is to be accurately taken; but with the above assumptions we have obtained formulae which are comparatively simple, are straightforward to compute, and are exact solutions. We also provide tables of the running efficiency, i.e. the percentage of running time to total (stopped + running) time; hence decisions can be made (by interpolation if necessary) with

SUMMARY The effects on the distribution of least-squares residuals of a series of model mis-specifications are considered. It is shown that for a variety of specification errors the distributions of the least-squares residuals are normal, but with non-zero means. An alternative predictor of the disturbance vector is used in developing four procedures for testing for the presence of specification error. The specification errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity. THE objectives of this paper are two. The first is to derive the distributions of the classical linear least-squares residuals under a variety of specification errors. The errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity. It is assumed that the disturbance terms are independently and normally distributed. It will be shown that the effect of the specification errors considered above is, with the exception of the error of heteroskedasticity, to yield residuals which though normally distributed do not have zero means, so that the distribution of the squared residuals is non-central x2. The second objective is to derive procedures to test for the presence of the specification errors considered in the first part of the paper. The tests are developed by comparing the distribution of residuals under the hypothesis that the specification of the model is correct to the distribution of the residuals yielded under the alternative hypothesis that there is a specification error of one of the types considered in the first part of the paper. As a preliminary step to deriving the test procedures the classical least-squares residual vector is transformed to a sub-vector which has more desirable properties for testing the null hypothesis that the specification of the model is correct. Also, under certain assumptions, with respect to the alternative hypothesis, it is shown that the mean vector of the residuals can be approximated by a linear sum of vectors qj

IN a previous paper the estimation of the sum X + ,> of the parameters of a simple birth and death process was considered. In the present paper we consider the estimation of X(i + ut)-1. This problem is equivalent to the problem of estimating the probabilities of steps to the right and left from an observed realization of a random walk which has one absorbing boundary and which is terminated, if necessary, after a preassigned number of steps. The properties of various estimators are considered.

SUMMARY A study is made of the fitting of discrete data by linear regression planes and by polynomials according to the criterion of least total absolute deviations. A form of solution is proposed which makes use of the simplex method of linear programming. An example is given to show the practical application of the method. THIS paper is concerned with the smoothing of discrete data by fitting functions which are linear in their parameters, such as polynomials and regression planes. The usual method adopted is the principle of least squares. As is well known, for observations having errors which are uncorrelated and which have constant variance, this method gives unbiased, minimum-variance estimates of the linear parameters, and accords with the maximum likelihood principle when the errors of observation are normally distributed. An alternative and sometimes preferable method would be to minimize the total sum of the absolute deviations of the observations from the approximating function. This method, which was first advocated by Edgeworth (see Bowley, 1928) has been largely neglected in the literature on account of the much greater labour of computation, but with the introduction of the high-speed digital computer this objection loses much of its force. It has been shown by Stiefel (1960) that there is a close connection between Chebyshev approximation (in which the maximum departure is minimized) and the simplex method of linear programming. The purpose of the present paper is to develop a similar correspondence for Edgeworth approximation. 2. COMPARISON WITH LEAST SQUARES

SUMMARY This paper presents necessary and sufficient conditions under which a random variable X may be generated as the ratio ZI V where Z and V are independent and Z has a standard normal distribution. This representation is useful in Monte Carlo calculations. It is established that when 7 V2 is exponential, X is double exponential; and that when WV has the asymptotic distribution of the Kolmogorov distance statistic, X is logistic.

It is well known that in the Gauss-Markov model (Y, Xβ, σ 2 V) with |V| ≠ 0, the BLUE (best linear unbiased estimator) of Xβ is Y 1 , the orthogonal projection of Y on M(X), the space spanned by the columns of X, with inner product defined as (x, y)=x'V −1 y. A quadratic function of Y 2 , the projection of Y on the orthogonal complement of M(X), provides an estimate of σ 2 . It may be seen that Y=Y 1 +Y 2 . When V is singular, the inner product definition as in non-singular case is not possible. In this paper a suitable theory of projection operators is developed for the case |V|=0, and a decomposition Y=Y 1 +Y 2 is obtained such that Y 1 is the BLUE of Xβ and a quadratic function of Y 2 is the MINQUE (Minimum Norm Quadratic Unbiased Estimator) of σ 2 in the sense of Rao (1972).

Let Xt (t = 1, 2, ..., N) denote a series of independent normal random variables, with zero means and non-zero finite variances ft (t = 1, 2, ..., N). Then ft and ft 1 admit finite Fourier expansions. The present paper is a study of the statistics N-1 S Xt cos (2tts/N) and Nt t Xt sin (2'rrts/N) as estimators of the sth Fourier coefficients of ft, and of the distributional properties of suitable ratios of these statistics as (sufficient) test statistics for non-constancy of ft, directed at alternatives where the Fourier expansion of ft 1 has a simple form.

FOR populations arranged in natural order, say in increasing values of a concomitant variable, one common sampling scheme is to divide the population into strata and sample proportionately from each stratum. Variance of the sample mean is minimized (with the possible exception of finite corrections) if the population is divided into n strata and one unit selected from each. A second common procedure, particularly if the sampling is with unequal probabilities, is to sample systematically.t It is well known that if the y characteristic is composed of a linear trend plus random elements the 1-per-stratum design is more efficient than systematic sampling of the population in natural order. The disadvantage of both of these sampling schemes is, of course, that no unbiased estimator of variance is available. In this paper we develop a sampling procedure which for n > 4 and a linear trend has a smaller variance for the sample mean than 1 per stratum and for which an unbiased estimator of the variance is available.

GIVEN is a finite population of N units with characteristics Yt (t = 1,2, ..., N) whose total Y = Yi +Y2 + . .. +YN is to be estimated. If a sample of size n is to be drawn from such a population, it is often advantageous to select the units with unequal probability. For example, such a procedure may be useful when measures of sizes xt are known for all N units in the population which are positively correlated with the characteristics yt. In such cases, one may utilize the knowledge of the xt by selecting units with probabilities proportional to sizes xt, although this is, of course, not the only way of using the known xt. Of the literature on sampling with unequal probabilities and without replacement we mention papers by Horvitz and Thompson (1952), Narain (1951), Yates and Grundy (1953), Des Raj (1956) and Hartley and Rao (1962). There are some limitations, of varying importance, attached to all these methods. Briefly speaking, the method of Horvitz and Thompson (1952) is applicable only under severe restrictions on the prescribed probabilities, the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956) require a cumbersome evaluation of working probabilities, and Hartley and Rao (1962) give only asymptotic variance formulae for the estimates of Y for large and moderate size populations N. The present method is an attempt to avoid all these disadvantages at the expense of a slight loss in efficiency. It has the following properties: (i) It permits the computation of an estimator of the population total which has always a smaller variance than the standard estimator in sampling with unequal probabilities and with replacement. (ii) Unlike the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956), the present method does not entail heavy computations, even for sample size n > 2, for drawing the sample or computation of the estimator and its variance estimate.

The system developed in the first paper of this series for the classification of states as open- or closed-shed type is applied to atomic states. These may be classified in the isoelectronic limit (Z ∞) from knowledge of the true wave function in this limit. One-matrix occupation numbers are tabulated for a number of states of the first-row atoms in the limit Z ∞ and the states classified. A classification for finite Z is possible in the framework of the Z-dependent perturbation theory by use of a thoerem for the stability of a closed-shed with respect to small perturbations. Such a stability does not hold in general for open-shel states. Dans le premier article de cette serie nous avons developpe une classification rigoureuse d'etats de type couche complete et incomplete. Dans le present article nous appliquons cette methode aux etats atomiques, qui peuvent etre classifies dans la limite Z ∞ des series isoelectroniques, parce qu'on en connait la fonction d'onde exacte. Nous donnons des nombres d'occupation de la matrice-densite du premier ordre pour un nombre d'etats des atomes de la premiere ligne dans la limite Z ∞, et nous classifions les etats correspondants. Pour Z fini on peut definir une classification dans le cadre de la theorie des perturbations grâce a un theoreme sur la stabilite d'un etat a couches completes en fonction de petites perturbations. Pour les etats a couches incompletes il n'existe en general pas de stabilite correspondante. Die strengen definitionen der Zustande mit abgeschlossenen und offenen Schalen, die im ersten Artikel dieser Reihe eingefuhrt worden sind, wurden auf Atomzustande angewendet. Diese konnen in der Grenze Z ∞ der isoelektronischen Reihen klassifiziert werden, weil die exakte Wellenfunktion in dieser Grenze bekannt ist. Besetzungszahlen der Einteilchendichtematrix werden fur eine Reihe von Zustanden der Atome der ersten Zeile in der Grenze Z ∞ gegeben und die entsprechenden Zustande werden klassifiziert. Fur endliche Z ist eine Klassifikation fur Zustande mit abgeschlossenen Schalen moglich im Rahmen einer Storungstheorie, wobei ein Satz uber der Stabilitat eines Zustands mit abgeschlossenen Schalen mit Rucksicht auf kleine Storungen benutzt wird. Fur Zustande mit offenen Schalen gibt es im allgemeinen keine solche Stabilitat.

Let F n (x) be the sample distribution function derived from a sample of independent uniform (0, 1) variables. The paper is mainly concerned with the orthogonal representation of the Cramer-von Mises statistic W 2 n in the form Σ ∞ j=1 (jπ) -2 z 2 nj where the z nj are the principal components of $\sqrt n\{F_n(x) - x\}$ . It is shown that the z nj are identically distributed for each n and their significance points are tabulated. Their use for testing goodness of fit is discussed and their asymptotic powers are compared with those of W 2 n , Anderson and Darling's statistic A 2 n and Watson's U 2 n against shifts of mean and variance in a normal distribution. The asymptotic significance points of the residual statistic W 2 n - Σ p j=1 (jπ) -2 z 2 nj are also given for various p. It is shown that the components analogous to z nj for A 2 n are the Legendre polynomial components introduced by Neyman as the basis for his "smooth" test of goodness of fit. The relationship of the components to a Fourier series analysis of F n (x) - x is discussed. An alternative set of components derived from Pyke's modification of the sample distribution function is considered. Tests based on the components z nj are applied to data on coal-mining disasters.

S(H: E fG) = h{P(H. E. G), P(E. G), P(H)}-h{P(G. H), P(G), P(H)} and the proof of Theorem 6 is invalidated. A valid proof can, however, be constructed without undue difficulty and the details will be supplied on request. Professor Aczel and I believe that, by using methods of Aczel (1966), the references to differentiability in my paper can be replaced by references to continuity, provided that the statement of Theorem 16 and of (vi) on p. 330 are changed to

[Read at a RESEARCH METHODS MEETING of the SOCIETY, April 8th, 1964, Professor D. V. LINDLEY in the Chair] SUMMARY In the analysis of data it is often assumed that observations Yl, Y2, *-, Yn are independently normally distributed with constant variance and with expectations specified by a model linear in a set of parameters 0. In this paper we make the less restrictive assumption that such a normal, homoscedastic, linear model is appropriate after some suitable transformation has been applied to the y's. Inferences about the transformation and about the parameters of the linear model are made by computing the likelihood function and the relevant posterior distribution. The contributions of normality, homoscedasticity and additivity to the transformation are separated. The relation of the present methods to earlier procedures for finding transformations is discussed. The methods are illustrated with examples.

THE object of this paper is to propose for discussion the following topic: b1, b2, ... are unbiased estimates of P1, ,2, ... , distributed normally with variances and covariances jointly estimated, with f degrees of freedom and independently of bl, b2, . .. , as vll, v12, V22, . .. , and the functions F,(cc) do not involve the parameters P3, What can we say about the roots of the equation in F(r, M) = rlFl (a) + P2 F2 (a) + . . . = 0? Numerical examples are discussed in detail to illustrate the problems of determining the fiducial distributions of (i) the root of a simple equation (i.e., a ratio), (ii) the roots of a quadratic equation with variable coefficients. The solutions proposed are based on a consideration of the region of the (aC, t2) plane lying above the curve

SUMMARY In this paper a branch of Bayesian statistical decision theory is investigated. A wide range of prediction problems in regression situations is considered within one statistical framework. A model is derived and applied to problems of classification.

SUMMARY Care needs to be exercised in the choice of model for a mixture system. The polynomial model, for example, cannot satisfactorily account for components which are inert or have additive effects and its coefficients lose their interpretative value when the variables are the proportions of components in the mixture. This paper proposes models constructed from functions homogeneous of degree one which overcome these difficulties, yet leave a wide choice of models.

SUMMARY We observe a record consisting of a "signal" plus "noise" up to the time instant t, and wish to extract the value of the signal at the time instant (t + m) by means of a linear filter. Here, m may be positive (corresponding to "prediction") or negative (corresponding to "smoothing"). In this paper we consider the case where both the signal and noise processes are nonstationary and possess evolutionary spectral representations. Using this approach we obtain a close analogue of the Wiener-Kolmogorov treatment of the stationary case, and show that the optimum filter is uniquely determined by the form of the evolutionary spectra.

SUMMARY Two general models for a queue in which groups of entities arrive at a single service line and are serviced in groups are defined. Various equilibrium properties for both models are established in terms of the traffic intensity p. For the special case of Poisson arrivals the first model is analyzed with reference to the imbedded Markov chain, the waiting time, and the busy period. It is demonstrated that if the entities arrive in groups the stationary distribution of the imbedded Markov chain does not agree with the general equilibrium distribution obtained by letting time t -> oo. For the special case of exponential service the stationary distribution of the imbedded Markov chain for the second model is obtained and the waiting time problem is discussed briefly. 1. GENERAL DEFINITION OF A BULK QUEUE The first treatment of a group queue occurred in the work of N. T. J. Bailey (1954) and F. Downton (1955). In their model the customers still arrived singly but were serviced in groups of s. Bailey derived the stationary distribution of the number of customers in the queue, and Downton obtained the Laplace transform of the waiting time distribution in equilibrium. In this paper an attempt is made to treat the general problem in which the customers arrive in groups and are serviced in groups. Since both Bailey and Downton referred to their group service as bulk service, this general model will be given the title of "bulk queue". The general model for a bulk queue can be described as follows. Groups of entities (individuals or elements) arrive at a service line and are serviced in groups; the service groups do not necessarily coincide with the arrival groups. The number of entities in an arrival group and the length of time to the arrival of the next group (inter-arrival time) are independent random variables which are identically (respectively) distributed with each succeeding group. Similarly, the number of entities allowed in a service group, and the service period time for that group, are independently and identically (respectively) distributed with each succeeding service group. The service random variables are independent of the arrival random variables.

SUMMARY In this paper the moment generating function (m.g.f.) of the truncated n-dimensional normal distribution is obtained. From the m.g.f., formulae for E(Xi) and E(Xi Xj) are derived, and are used to investigate certain special cases. Some applications of these results to statistical genetics are also discussed.

SUMMARY The multivariate normal distribution can be used to describe the response variable of a system. A more comprehensive multivariate model is described in this paper: it has a distribution describing an error variable internal to a system, with a known multivariate distribution; and it has a positive affine transformation, the physical quantity, which generates a response vector from an error vector. This more comprehensive model is a structural model and it provides structural probability statements concerning the physical quantity. Error and structural distributions are derived for the multivariate model. The structural distribution for a quantity can be used to generate structural prediction distributions: various prediction distributions are obtained for the multivariate model. The results are specialized to cover the case of the multivariate normal structural model. The classical multivariate normal has been analysed by Bayesian methods (Geisser and Cornfield, 1963). The more comprehensive multivariate

The problem of obtaining consistent estimates for structural parameters in the presence of infinitely many incidental parameters was discussed first by Neyman and Scott (1948). In this paper a maximum-likelihood method based on the conditional distribution given minimal sufficient statistics for the incidental parameters is suggested. It is proved that conditional maximumlikelihood estimates in the regular case are consistent and asymptotically normally distributed with a simple asymptotic variance. The efficiency problem of this new estimator is discussed in particular with respect to some situations with ancillary information.

SUMMARY This paper examines the various classical results in the theory of regression and double sampling estimation and extends them to the study of a finite population.

During the last few years, several authors have devoted attention to both old and new methods of making inferences from likelihood functions. In this paper we study the properties of interval estimates which are obtained by drawing a horizontal line across the graph of the likelihood function. An exact evaluation is reported for one case of positive and one of negative binomial sampling, with parameter 0. The attained confidence coefficient cx(O) is graphed over 0? 0 < 1 for these two cases. A brief look is taken at intervals obtained when the random variable is continuous.

SUMMARY THE minimization of a convex function of variables subject to linear inequalities is discussed briefly in general terms. Dantzig's Simplex Method is extended to yield finite algorithms for minimizing either a convex quadratic function or the sum of the t largest of a set of linear functions, and the solution of a generalization of the latter problem is indicated. In the last two sections a form of linear programming with random variables as coefficients is described, and shown to involve the minimization of a convex ftunction. Linear programming has been studied extensively in the last few years, as indicated by Vajda (1955). Various authors have mentioned the possibility of relaxing the requirement of linearity, but the practical problems of non-linear programming do not seem to have been considered in any detail.* This paper is concerned with some aspects of the simplest form of non-linear programming-the minimization of a convex function of variables subject to linear inequalities. In principle this can always be done using the method of steepest descents, but this will rarely be practical in its primitive form. We therefore consider special methods for some important particular classes of such functions. In Section 2 the Simplex Method, originally developed by Dantzig (1951) for linear programming, is outlined in terms sufficiently general to cover the applications to non-linear programming considered in the next two sections. In Section 3 we show how to minimize a convex quadratic function. This enables one to use what.amounts to the Newton-Raphson Method for minimizing a well-behaved general convex function: one finds a feasible solution of the constraints, and at each stage minimizes the quadratic function whose first and second derivatives at the feasible solution are the same as those of the given function.

SUMMARY This paper gives modifications of eleven statistics, usually used for goodness of fit, so as to dispense with the usual tables of percentage points. Some test situations are illustrated, and formulae given for calculating significance levels.

SUMMARY A NUMBER of alternative probability models are considered for the interpretation of failure data when there are two or more types of failure. Some of the statistical techniques that can be used for such data are illustrated on an example discussed recently by Mendenhall and Hader. Mendenhall and Hader (1958) have recently given an interesting account of a model for the analysis of failure-time distributions when there are two, or more, types of failure. They illustrate their theory by analysing some data on the failure-times of radio transmitter receivers; the failures were classed into two types, those confirmed on arrival at the maintenance centre and those unconfirmed. In the present paper their example is used to illustrate and distinguish between a number of models that can be used for this type of data. The essential feature of the problem is that we have independent individuals exposed to risk, and that on failure an individual is withdrawn from risk. We observe, for example, that individual number one fails after life-time t, and that the failure is say of the first type: this means that we know the time at which failure of the first type occurs, but only that failure of the second and other types had not occurred by time t. In many applications, including Mendenhall and Hader's, the sample contains individuals that have not failed at the end of the period of the observation. Thus in their example no receivers were operated after 630 hr. Data like this arise in several fields in addition to industrial life-testing. For example, in medical and actuarial work the estimation and comparison of death rates from a particular cause requires corrections for deaths from other causes. In particular Seal (1954) and Elveback (1958) have discussed the more theoretical aspects of this in connection with actuarial work and given numerous references. Sampford (1954) has dealt with similar problems in bioassays. In tensile strength testing there may be two or more types of failure, for example jaw breaks and fractures in the centre of the test specimen; here the observation is load on failure, not life on failure. A further interesting application is in experimental psychology. Audley (1957) has interpreted latency measurements in learning experiments by postulating independent Poisson processes of A-responses and B-responses; the first process for which an event occurs is considered to determine the nature (A or B) of the response and the time at which it occurs.

It is the aim of the present paper to give a mathematically oriented foundation of BW-perturbation theory, which is along the lines of Kato's previous work for RS-perturbation theory. For this purpose we firstly derive the expressions of BW-perturbation theory by the use of the contour integral method (Kap. I). In Kap. II sufficient criteria for the convergence of BW-perturbation theory are derived and applied to the 1/Z-expansion of the isoelectronic series of the He atom. The characteristic differences of the derivation and convergence properties of the two different kinds of perturbation theory are discussed in detail.

SUMMARY The logistic process is a simple stochastic process often used to represent the spread of an epidemic. Some recent studies (Bailey, 1950; Haskey, 1954) provided information regarding the stochastic epidemic curve for this process. In this paper, we present much more extensive tables of this epidemic curve, and we note some potential uses for the logistic process in the behavioural sciences. 1. THE LoGISTIC PROCESS To describe this process, it is convenient to take as an illustration the spread of a disease. Suppose that n persons form a closed group and that a person infected by the disease is introduced into the group at time t = 0. Suppose that the infection spreads through contact among the members and that the probability of one new infection between time t and t+A, given that r members are uninfected at time t, is

SUMMARY In a recent paper (Welch and Peers, 1963) formulae were obtained for confidence points depending on the distributional properties of certain integrals of weighted likelihoods. Some comparisons are made here with the SUPPOSE a sample S has probability element p(S, 0) dS = exp {L(S, 0)} dS. We are concerned in the main with situations where L(S, 0) is "in the probability sense" 0(n), where n can become large. For "almost all samples" there is then a single maximumlikelihood estimate T which will differ from 0 by 0(n-A). We write K2(0) = E{(n-iaL/a6)2} = _E(n-E 32L/62), (1) but for convenience we shall often abbreviate K2(T) to K2 and K2(6) simply to K2. Asymptotically ni K'(T- 0) tends to be distributed normally with mean zero and standard deviation unity. The quantity n iK2(6 - T) tends also to have the same distribution. More precisely Pr{n K2(6-T) < x} = 0s+0(n-), (2) where e and cx are related by

1. The purpose of this paper is to give simple proofs for some recent versions of Linnik's large sieve, and some applications.The first theme of the large sieve is that an arbitrary set of Z integers in an interval of length N must be well distributed among most of the residue classes modulo p, for most small primes p, unless Z is small compared with N. Following improvements on Linnik's original result [1] by Renyi [2] and by Roth [3], Bombieri [4] recently proved the following inequality: Denote by Z(a, p) the number of integers in the set which are congruent to a modulo p.

SUMMARY Scheffe, in two recent papers (1958, 1963), has given designs for experimenting with mixtures. The basis of these designs is the choice of symmetrical arrangements of points in the factor space and the fitting of carefully chosen models which have exactly as many coefficients as there are data points. The designs are such that some of the experiments do not contain any of one or more ingredients of the mixture. This may or may not be a disadvantage depending on the problem involved. Here an alternative form of design selection is made for the case of mixtures of three factors. (The method has also been extended to four factors.) This involves the premise that, in the absence of specialist knowledge about the form of the true response function, it is desired to fit a response surface equation of first or second order over the factor space of possible mixtures, and experimental runs are needed which, in a certain sense, ensure the best surface fit possible. The principles used in the choice of appropriate designs will be those originally introduced by Box and Draper (1959).

SUMMARY For a distribution depending on a single parameter the first four sampling moments of the maximum-likelihood estimate to orders N-2, N-3, N-3 and N-4 respectively are given. Expressions for the measures of skewness Y1 and y2 are also given. Several illustrative examples are included as a check on the heavy algebra. The paper extends earlier work by Haldane and Smith.

SUMMARY This paper is concerned with the problem of testing the existence of a trend in the means Gi of Poisson distributions. It is assumed that these means are changing exponentially, that is, log Gi = ci+/x2. A classical method is reviewed which is used for testing the hypothesis P = 0. The exact Bayesian distribution for P is derived and a Bayesian approximation suggested which proved to be very useful. Finally, a comparison of these three methods by means of numerical examples is made.

Professor R. L. PLACKETT in the Chair] SUMMARY This paper is concerned with the analysis of data from a multiple regression of a single variable, y, on a set of independent variables, xl, x2, .. . ,xr. It is argued that the form of the analysis should depend on the use that is to be made of the regression, and that therefore an approach employing ideas from decision theory may be worth while. Two situations are analysed in this way: in the first it is desired to predict a future value of y; in the second we wish to control y at a preassigned value. The two analyses are found to be different: in particular, the standard errors of the regression coefficients are found to be irrelevant in the prediction problem, but not in the control problem. In the former it is shown that, under rather special assumptions on the multiple regression experiment, the analysis is similar to that recommended by other writers. If the costs of control do not depend on the values at which the control takes place, a similar analysis holds for the second problem. The approach throughout is Bayesian: there is no discussion of this point, I merely ask the non-Bayesian reader to examine the results and consider whether they provide sensible and practical answers.

SUMMARY An ergodic theory is developed for the subadditive processes introduced by Hammersley and Welsh (1965) in their study of percolation theory. This is a complete generalization of the classical law of large numbers for stationary sequences. 1. SUBADDITIVE PROCESSES IN an important paper Hammersley and Welsh (1965) introduced the concept of a subadditive stochastic process, and they have shown how such processes arise naturally in various contexts, but particularly in the study of random flows in lattices. They have shown that one may expect these processes to exhibit a certain ergodic behaviour, and have taken the first steps towards the construction of an ergodic theory like the classical one for averages of stationary sequences. If T is any subset of the real line, a subadditive process x on T is a collection of (real) random variables xst(s, t E T, s < t) with the property that

SUMMARY The method of minimum necessary samples formulated by Novick and Hall from a suggestion by Lindley is refined and extended to cover multiparameter models. A number of such models are studied and Bayesian indifference specifications are obtained for each within extended natural conjugate families. In all cases the one-parameter conditional specifications derived from the multiparameter specifications are consistent with specifications derivable from corresponding one-parameter models. Models studied include the multinomial models, one- and two-parameter uniform models, an exponential model with a range parameter, and the two-parameter normal model. For these models the specifications seem entirely straightforward. Indifference specifications are also obtained for the simple regression model and the bivariate normal model. It is noted that these specifications do not agree but the relationship between these two specifications is described and in part justified. It is pointed out that these differences could, in fact, mirror important differences in the structural assumptions of these models. A random effects model of value in mental test theory is studied briefly. This model provides a framework for a discussion of the multiple comparisons problem. Finally some summary comments on the methods of this paper are offered.

SUMMARY In testing differences between proportions or rates of occurrence of events in different categories, expected numbers may be calculated separately within certain strata and subsequently summed over strata. For the Poisson case an approximate X2-statistic is derived as a simple modification of the usual test statistic. Other approaches and situations are discussed more briefly. IN many situations in which a number of proportions are to be compared the familiar x2-heterogeneity test needs to be modified to take account of some natural blocking or stratification of the observations. For example, to compare the proportions of individuals in m occupational groups who exhibit certain symptoms, one could apply the usual X2-test on m -1 degrees of freedom. It would, however, be sensible to take account of the age distributions of the individuals in the m groups because the prevalence of symptoms may be markedly affected by age. The individuals may therefore be subdivided into 1 age groups. A X2-test on mr-1 degrees of freedom could be carried out in each age group, but it would be useful to have a composite test, also on m -1 degrees of freedom, to assess the systematic differences between occupational groups within age groups. In vital statistics such problems are traditionally handled by methods of standardization, and these methods are widely used also in epidemiological survey work in problems similar to that referred to above. Standardization is, however, often performed without reference to sampling variation. In this paper I discuss methods of obtaining an asymptotically valid X2-test on mr-I degrees of freedom, with particular reference to the method of indirect standardization. Suppose that the main comparison is between m categories, denoting a typical category as the jth, and that the secondary subdivision is into 1 strata, of which a typical one is the ith. The observed proportion in the (i,j)th cell is N?>/t?, where Ni, is the number offailures and tij is the number of cases. The notation is shown in Table 1, in which marginal totals are denoted in the usual way except that we use n and t rather than n and t.. The distinction between capital and lower-case letters is adopted intentionally because in the statistical treatment only the Nij and Nj are regarded as random variables. In many epidemiological and vital statistical applications the NP4 may be regarded, apart from linear restraints, as Poisson rather than binomial variables. This may be either because the basic situation is as shown in Table 1, with all the failure-rates

SUMMARY The paper is concerned with the lengths of intervals in a stationary point process. Relations are given between the various probability functions, and moments are considered. Two different random variables are introduced for the lengths of intervals, according to whether the measurement is made from an arbitrary event or beginning at an arbitrary time, and their properties are compared. In particular, new properties are derived for the correlation coefficients between the lengths of successive intervals. Examples are given. A theorem is proved, giving conditions under which two independent stationary point processes with independent intervals may be superposed, giving a new point process which also has independent intervals. Mention is made of the application to the theory of binary random processes and to the zeros of a Gaussian process.