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This paper proposes resolving the question of non-uniqueness in ancillary statistics by taking the ancillary statistic which has the smallest entropy. It also gives an example of non-uniqeness of ancillary statistics in which all the objective resolutions fail to provide a unique ancillary statistic.

In this article we derive the exact finite sample distribution of a statistic utilized by Theil [10] to test whether sample and prior information are in conflict with each other. A weighted estimator for the slope coefficient in a simple regression equation with one independent variable is obtained by using weights which are determined by the observed value of Theil's test statistic. This weighted estimator is particularly useful when we are uncertain about the compatibility of prior information with sample information.

A case of nonregular estimation arises in attempting to estimate a single unknown parameter, $\theta$, in the probability distribution of a single chance variable in which one or both of the extremities of the range of the distribution are functions of the unknown parameter. The case treated in this paper is the one in which a probability density of exponential type exists. When one extremity alone of the range depends non-trivially upon $\theta$, a necessary and sufficient condition is given in order that a single order statistic be a sufficient statistic for $\theta$. In this case conditions are given for the existence of a unique unbiased estimate of $\theta$ possessing minimum variance uniformly in $\theta$. In the case in which both extremities of the range depend upon $\theta$, a necessary and sufficient condition is given that the smallest and largest order statistics constitute a set of sufficient statistics for $\theta$. In this case Pitman [1] has shown that a single sufficient statistic exists if one extremity of the range is a monotone decreasing function of the other extremity. It is shown that under the above condition a unique unbiased estimate exists possessing minimum variance. Moreover a surmise of Pitman is proved that only under this condition does a single sufficient statistic exist. When a single sufficient statistic does not exist, an unbiased estimate of a known function of $\theta$ is obtained which has less variance than any analytic function of the set of sufficient statistics for $\theta$.

SUMMARY A modified form of the Pearson x2 test statistic is considered where estimators based on the ungrouped sample are employed in the test statistic as well as in determining the class interval end points. In the case of continuous distributions with location and scale parameters, it is shown here that the null distribution of such a modified test statistic does not depend on the parameters if these are estimated by sample mean and variance. By utilizing known results concerning the distribution of a weighted sum of independent chi-squared variates, a table of certain percentage points of the asymptotic distribution of this modified test statistic is developed in order to facilitate its use for testing normality. case of continuous distributions. In the first place, how should the class intervals be formed and how many should there be? Secondly, if there are unknown parameters, how should they be estimated and what is their effect on the test? Aside from the complexities arising in deriving estimators ofthe parameters, the resulting distribution ofX2, the Pearson x2 statistic, can be quite different from that of x2, depending upon the method of estimation, as was observed by Chernoff & Lehmann (1954). Thirdly, the estimation of class probabilities when parameters are unknown is also usually complicated in the case of continuous dis- tributions. In considering the problem of how the class intervals should be formed in using the X2 statistic, A.R. Roy in an unpublished report, and Watson (1957, 1958, 1959) relaxed the requirement of fixed class boundaries and utilized estimators of the parameters to determine the end points of the intervals. The estimators employed for this purpose as well as for substitution in the X2 statistic were based on the ungrouped sample. In the present paper we extend a result of Roy to show, in the case of unknown location and scale parameters, that the asymptotic null distribution of the test statistic does not involve the unknown parameters when they are estimated by the sample mean and sample variance respectively. As an application of the technique developed here, we provide a table of some percentage points of the asymptotic distribution of the Pearson statistic modified in the above manner, based on equal probability classes, and used to test for normality. The number of class intervals to be employed in the test should depend on the alternative distributions. This requires knowledge of the corresponding asymptotic nonnull distribution of the test statistic and will be given for various alternatives in a subsequent

A general numerical procedure is given for obtaining the percentage points of an arbitrary statistic, given only the distribution of the individual data points.

Complete sufficient statistics and minimum variance unbiased estimators for parameters of several useful families of distributions are listed. These families are identified as having or not having the property of competeness

The Mann-Whitney-Wilcoxon statistic is of some importance in non-parametric statistical inference. The asymptotic normality of this statistic has been established previously for the case where the distributions of the random variables, in each of the two samples, are the same. In the present work it is shown that this statistic is asymptotically normal when these distributions are not all the same. An application of this result to a particular hypothesis testing problem is also given.

The problem of finding a Chebyshev type inequality for random variables with unknown or non-existent variance was considered by Z. W. Birnbaum (1970). In this present paper, a statistic, $T$, similar to, but simpler than Birnbaum's, is considered. The statistic is independent of location and scale parameters for families of bell-shaped distributions and so may be considered to be a competitor to Student's $t$. An inequality establishing an upper bound for $P(|T| > \lambda)$ is proved. This bound is considerably smaller than the corresponding bound found by Birnbaum. Finally, an improvement of the latter is offered.

1. Introduction. In 1960, Bahadur [3] proposed two measures of the asymptotic performance of tests: one approximate measure, based on the limiting distribution of the test statistic; and one exact, based on the limiting form of the probability of a large deviation of the statistic from its asymptotic mean. Although knowledge of the exact measure is more desirable, it is often difficult to compute, whereas the approximate measure is usually trivially available. It is to be hoped that the approximate measure will nearly always be a good approximation to the exact measure, and there is considerable evidence in support of this conjecture, but counterexamples do exist (see e.g. [1], p. 20). This paper explores the question for the Kolmogorov-Smirnov (K-S) and Kuiper statistics-two closely related measures of goodness-of-fit based on deviations of the sample distribution functions from the null case-in one- and two-sample situations. In the case of the weighted one-sample K-S statistic and the two-sample Kuiper statistic, it is possible to obtain exact measures although the corresponding approximate measures are not available. The relative efficiency (in a sense defined in Section 2) of the weighted one-sample K-S statistic to the unweighted K-S statistic is computed in a number of cases, and the relative efficiency of the Kuiper statistic to the unweighted K-S statistic is also examined, from which it appears that the Kuiper statistic is always at least as good (and often much better) than the K-S statistic. In the cause of brevity, a great deal of the theory and all the computational detail (generally very tedious) has been omitted. Much of it is available in [1]. However, a short resum6 of the theory of the Bahadur measures of performance and their properties is given in Section 2 for the sake of completeness, although the details here are also omitted.

In “Desisjonsteorien og Neyman-Pearson teoriens stilling idag”, Sverdrup [3] considers in section II, C, D and E, conditional test methods, and he discusses the well-known fact that the principle of unbiasedness in certain cases implies conditional tests. Sverdrup doubts (illustrated by examples), that the principle of unbiasedness always is the real reason for choosing conditional tests. He suggests that the ancillary principle, introduced by Fisher [2], might be more basic. We exhibit in section 3 of the present paper an example which shows that one can get poor tests if one conditions on a statistic which is ancillary in the (perhaps) commonly accepted sense of this term. We analyze the reason for this shortcoming, and we propose that the usual explanation of ancillary statistics should be supplemented, resulting in a definition in section 3 below.

Joint intervals of bounded confidence are suggested for the medians of a bivariate population with continuous marginal distributions. The two intervals are of the classic type based on sample order statistics.

Summary In this note a problem on exact moments of order statistics from a power-function distribution is considered. The characteristic function of the kth order statistic is obtained and moments about the origin of the kth order statistic are expressed in terms of gamma functions. An exact expression for the covariance of any two order statistics Yi < Yj is obtained in terms of beta and gamma functions. Various recurrence relations between the expected values of order statistics are also obtained.

The distributions of midrange, half-range, median and geometric mean of samples from a triangular population are derived. Mean, median, and midrange are compared as to their efficiency in estimating the single parameter of the population, which is the upper end of the range of the distribution. The best estimator of the parameter, however, is based on the maximum value in the sample, since it is derived from the likelihood function and is a sufficient statistic.

Let the probability density of observations be denoted by φ(x | θ), where x stands for the variables and θ for the parameters. A function t of the observations is called an unbiased estimate of the function ψ(θ) of the parameters ifwhere dx stands for the product of differentials.

The “sum of chi-squares” technique suggested by several authors attempting to assess the fit of certain learning models to data (Atkinson, Bower and Crothers, 1965, Calfee and Atkinson, 1965) is examined in a general framework and its asymptotic distribution theory derived. The fit statistic used is seen not to have an asymptotic chi-square distribution except under very special circumstances. Its asymptotic mean and variance are found both when no parameters are estimated and when s parameters are estimated by minimizing the fit statistic. Bounds are found for the asymptotic mean and variance of this statistic. Some partial results are given regarding the direction and size of the error made in assuming the fit statistic has an approximate chi-square distribution with the “right” degrees of freedom for calculating significance levels. The estimate of the parameter found by minimizing the fit statistic is examined and compared to some of its possible competitors. Finally these theoretical results are applied to some data given by Calfee and Atkinson (1965).

This paper deals with the classification statistic Z due to John [4], which is a competitor to the Anderson statistic W in discriminant analysis with two multivariate normal populations. An asymptotic expansion of the distribution of Z as well as the associated probability of misclassification with respect to the three numbers of degrees of freedom is given. Some coefficients in the expansion for the latter are computed numerically and compared with those for W , which were given before by Okamoto.

For the two-sample problem, Wilcoxon [21], Fisher and Yates [6], Terry [19], Hoeffding [10], Hodges and Lehmann [8], Savage [17], Chernoff and Savage [3], Lehmann [13], Capon [2] and others have considered rank-sum statistics equivalent to $S_N(H) = m^{-1} \sum E(Z(R(X_i)) \mid H) - n^{-1} \sum E(Z(R(Y_i)) \mid H)$, where $E(Z(j) \mid H)$ is the expectation of the $j$th order statistic of a sample of size $N = m + n$ from a population with cpf (cumulative probability function) $H$ and $R(X_i) \lbrack R(Y_j)\rbrack$ is the rank of $X_i\lbrack Y_j\rbrack$ in the combined sample of $X$'s and $Y$'s. In order to perform tests based on these statistics one needs special tables of expected values as well as tables of the hypothesis distribution. Further, in general, exact desired significance levels can only be achieved through randomization. The object of this note is to introduce rank-sum statistics in which one randomizes initially and circumvents the necessity of two special tables. These new randomized statistics, which are formed by deleting the expectation signs "$E$" in $S_N(H)$, generally satisfy the same asymptotic goodness criteria as their nonrandomized counterparts. Moreover, they have the added advantage that for appropriate choices of the parameters they have null hypothesis distributions which are continuous, known and tabulated (e.g., normal, $\chi^2$, etc.) In particular, one of the new two-sample statistics has an exact normal distribution and is asymptotically uniformly more efficient than the $t$-test for translation alternatives. This idea is extended to obtain randomized rank-sum statistics for the independence, randomness, $k$-sample and two-factor problems analogous to the statistics of Friedman [7], Puri [15], Stuart [18] and others. As in the references listed above, prime interest will be in those cases for which $H$ is normal, uniform or exponential. However, the methodology is equally applicable to other continuous cpf's $H$.

This is an expository paper, pointing out explicitly the pseudoness of the “F-statistic” used in stepwise procedures for determining the independent variables to be used in a linear prediction equation. Unfortunately this pseudoness prevents one from obtaining any probabilistic measure of the goodness of the final prediction equation. The use of the distribution of an order statistic is discussed as an aid to understanding the problem as well as furnishing a “better” (although still unsatisfactory) approach.

A variety of statistics are available for testing the homoscedasticity of several populations. In this article a number of these statistics are compared for power and for stability of error rates. Bartlett's statistic is seen to be very powerful in all the experimental cases considered here. Under a condition of non-normality the only statistic to maintain stable error rates is the one using the logarithmic transformation with the analysis of variance.

In Part 1 (Stephens, 1963) were given the moments, and some small-sample results, of the distribution of UN. This is a goodness-of-fit statistic introduced by Watson (1961, 1962), and tests the null hypothesis that N observations come from a cumulative distribution function F(x). Throughout this paper the distribution of UN will refer to the distribution on the null hypothesis. UN is particularly useful when the observations are points on a circle; Watson has shown that its distribution is independent of F(x), and its value, in the circular case, is independent of the choice of origin. Suppose the observations, in increasing order, are x., x2, ..., XN, and let y. = F(xj), with y the mean of the yi. The value of UN2 may be calculated from N 2-1 21 2 1 UN=.z1(jSI 2N) N(Y2) 12N (1)

SUMMARY THis paper gives the sampling distribution of a sufficient statistic (which is also the maximum likelihood statistic) for the parameter of a distribution when the range of the distribution depends upon the parameter. Confidence intervals for the parameter are obtained, based on the distribution of the sufficient statistic.

This paper examines Fisher's example of the Mile problem as well as his example of a bivariate normal model which he uses to illustrate the method of making inference on an unknown parameter from the maximum likelihood estimator conditional on an ancillary statistic. It is seen that the result obtained by Fisher can be easily achieved by fraser's structural method which depends neither on the conception of ancillarity nor on the maximum likelihood estimator. The same conclusion is reached on obtaining conditional confidence intervals for the parameters of location models.

0. Summary. In 1949, M. Kac defined a statistic which appears useful for statistical problems arising in insurance, biology, and telephone engineering [2]. In those fields, the natural observation peiiod is a fixed time period during which a random number of observations would be obtained. The distribution of the number of observations is given by a Poisson distribution. Distributions of this statistic can be used to determine upper and lower confidence contours for an unkrnown distribution, or in testing a distribution hypothesis. The purpose of this note is to extend the authors' earlier results, [1], to the two-sided Kac statistic.

It is proved that if there exists a sufficient statistic for the estimation of an unknown parameter of a population, the frequency function of the population must be of a certain type.It is shown that some modification of previous theory of the intrinsic accuracy of statistics is necessary when the range of the population sampled is a function of the parameter to be estimated.Finally, the theory is extended to sufficient sets of statistics, i.e. sets of statistics which together contain all the information provided by a sample about an unknown parameter.

Generalizing earlier observations by Fisher and Hotelling, Tukey [1] showed that if a family of distributions admits a set of sufficient statistics, then the family obtained by truncation to a fixed set, or by a fixed selection, also admits the same set of sufficient statistics (this wording is Tukey's; we give a precise mathematical statement later). Tukey's proof assumed the relevant family of probability measures to be dominated by a fixed measure function and made use of the factorization theorem concerning sufficient statistics in this case. In the present short note we shall first re-prove Tukey's result without assuming domination (and, hence, without appealing to the factorization theorem). Then we shall show that, under general conditions, if a sufficient statistic has one or more of the properties of completeness, bounded completeness, or minimality, before truncation, then it preserves such after truncation. The treatment is on the lines of the abstract discussion of sufficient statistics given by Halmos and Savage [2]. We shall assume familiarity with the results given in this latter paper. For definitions of completeness, bounded completeness, and minimality, and for a discussion of the significance of these concepts we refer to Lehmann and Scheffe [3].

The concept of "functional sufficiency" or "f-sufficiency" for a class of density functions is introduced and conditions given under which it corresponds to sufficiency as defined by Halmos and Savage [2]. A minimal f-sufficient statistic is defined and shown to exist, and its construction is given. The minimal f-sufficient statistic for a class of densities for which the region of positive density varies with the parameter is shown to be equivalent to the combination of a "statistic of selection" and the minimal f-sufficient statistic for a class of densities for which the region of positive density is fixed. The construction of sufficient statistics in this latter case subject to certain conditions of regularity has been treated by Koopman [1]. If the parameter is a parameter of selection from a fixed distribution, then the statistic of selection is the minimal f-sufficient statistic. If in addition the regions of positive density are monotone and indexed monotonely by a real parameter, then the statistic of selection is sufficient according to the Halmos and Savage definition. Three examples are given to illustrate the results.

SUMMARY A consistency criterion is proposed for fiducial probability distributions: if a fiducial density function is used to modulate a prior density function, the resulting frequency function should be that obtained by a Bayesian analysis. For a real variable and a real parameter, the fulfilment of this criterion is equivalent to a condition examined by Lindley (1958) and it requires the parameter to be essentially a location parameter. For several variables and the same number of parameters, the criterion if applied to fiducial probability derived from a pivotal quantity produces a simple differential equation condition on the pivotal quantity. If one-dimensional fiducial distributions can be generated in an arbitrary direction from any point in the parameter space, then the fulfilment of the criterion requires the model to be of transformation-parameter form; for such models a variety of properties concerned with consistency and interpretation are available (Fraser, 1961). The fiducial method of inference was introduced by Fisher (1930) and has been discussed and developed in many of his subsequent papers. The development is primarily in terms of examples, with but little attention to precise governing principles. In this paper the method will be interpreted in terms of the pivotal quantities that are a common feature in the examples. A pivotal quantity is a function of a sufficient statistic and a parameter that has a fixed frequency distribution. For a single real variable and a single real parameter the distribution function is a pivotal quantity and it has the uniform distribution on (0, 1) regardless of the parameter value. In some statistical problems there may be a natural choice for the pivotal quantity based, perhaps on the physical origin of the problem. The quantity may then express a position, a probability or error position for an outcome relative to the parameter value. In other problems there may be a natural choice based on mathematical properties of the specification. In this paper a pivotal quantity will be assumed given as part of the statistical problem: the distribution of the pivotal variable will be interpreted as describing the error distribution or error pattern, and the pivotal equation as describing the way that the probability mass of the error pattern is applied to the sample space. The fiducial method as presented by Fisher involves a preliminary reduction to an exhaustive statistic. If the exhaustive statistic is of the same dimension as the parameter it is used as the basic variable in the pivotal quantity. If the exhaustive statistic is of higher dimension, then an ancillary statistic is needed such that conditionally the exhaustive statistic has the same dimension as the parameter and can be used as basic variable in the pivotal quantity.

The Durbin-Watson statistic is used for testing the first order serial correlation among residuals in a linear model. It is based on the residuals from a corresponding regression analysis. In this paper a generalization of the statistic which tests for higher order dependence among residuals is proposed. The paper gives a brief review of the Durbin-Watson theory and the construction of the corresponding significance tables based on Jacobi orthogonal polynomials and the beta density. The distribution of the Durbin-Watson statistic based on Tmhof's distribution of Quadratic forms is indicated as an alternative method of directly computing the distribution function of the Durbin-Watson statistic, as well as its generalization, Significance values for 5 percent level of significance for lags 2 to 4 are given in Table II.

When applying the Markov model, it is often assumed the transition matrix is stationary and of order one. This article considers the problem of applying the transformed likelihood ratio statistic and the contingency statistic to testing the assumptions of a particular order and of stationarity when the assumptions are not true. A Monte Carlo study of the small sample power of both these test statistics shows that if the true alpha level is kept equal for the two tests they have similar power. However, if the alpha level is set by reference to any standard table of the chi-square distribution the likelihood ratio statistic has larger power.