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Craig (1962) described an application of order statistics from a binomial distribution and gave an approximate expression for the mean value of the minimum of two binomial variables having the same probability of success and the same number of trials. In this note we suggest another approximation for the mean and also for the standard deviation of the minimum, using normal order statistics tabulated by Teichroew (1956).

IN this paper Haldane points out (p. 142) a difference between his results for the mean and variance of x2 in a 2 x n-fold contingency table when the expectation p is fixed and the results obtained by me in my paper (Annals of Eugenics, Vol. VII, part III, p. 211). The difference is that I have (n 1) throughout where Haldane has n. Haldane writes, "my own results would appear to be slightly more accurate than Cochran's", which might, I think, give the impression that both Haldane's results and mine are only approximations. In fact, both results are mathematically exact, the difference between them being one of definition of x2. My paper is almost entirely concerned with the distribution of x2 when the expectation p is not known. In the results which I gave for the distribution of x2 when p is known, I retained the term S (x -x-)2 in the numerator of x2 instead of S (x np)2, to facilitate comparison between this and my other results. Thus my x2 has (n 1) degrees of freedom, whereas Haldane's x2 has n degrees of freedom. Unfortunately I did not emphasize this point in the passage concerned, and as it may have appeared misleading to others besides Haldane, I welcome this opportunity of drawing attention to it. Haldane's x2 is, of course, the one which is normally appropriate in testing the departure from independence when the expectation is known.

SUMMARY The bias of several logit estimators and their corresponding variance estimators is investigated in small samples. Their use in quantal bioassay is similarly explored. The logit transformation has been suggested in the analysis of higher dimensional contingency tables, by Woolf (1954) and many others more recently, and also in estimating the parameters of the logistic function in the quantal bioassay problem (Berkson, 1944, 1953). Various modifications of the logit have been suggested by Berkson (1953), Haldane (1955), Anscombe (1956), Tukey, mentioned by Anscombe (1956), and Hitchcock (1962). Modifications of its usual variance estimator have been proposed by Haldane, Goodman (1964) and Gart (1966). More recently Goodman, in an unpublished paper, has derived several further modifications of his estimator. In this paper we present a numerical comparison of the bias of these estimators and give conditions under which one or the other may be preferred. The use of logits in the quantal bioassay problem is briefly explored with particular reference to the asymptotic results of Hitchcock regarding the bias of the estimators.

SUMMARY The effect of nonnormality on the Welch approximate degrees of freedom t test is demonstrated. A two-sample trimmed t statistic for unequal population variances is proposed and its performance is also evaluated in comparison to the Welch t test under normality and under long-tailed distributions. If the underlying distribution is long-tailed or contaminated with outliers, the trimmed t is strongly recommended. Some key word8: Behrens-Fisher problem; Robust test; t distribution; Trimming; Winsorization. Testing the equality of two means from independent samples is a common statistical problem. If the underlying distributions are normally distributed with equal population variances, it is well known that one would use the Student's t test. Unfortunately, this test statistic is sensitive to some nonnormal situations and that leads us to consider other more robust alternatives. Among the alternatives is the two-sample trimmed t, proposed and evaluated by Yuen & Dixon (1973); definitions of trimming will be given later. This statistic can be easily computed and its distribution is satisfactorily approximated by that of a Student's t with the degrees of freedom corresponding to the reduced sample. Results show that the loss of power efficiency in using trimmed t is small under exact normality, while the gain may be appreciable for long-tailed distributions. In cases where distributions are normal but population variances are unknown, the first exact solution was given by Behrens (1929) and later extended by Fisher (1939) as the fiducial solution. Among many others who have worked on this problem, Welch (1938, 1949) provided an approximate degrees of freedom t solution to his asymptotic series solution. Wang (1971) studied the probabilities of the type I errors of these two Welch solutions for selected cases and concluded that in practice, one can use the Welch approximate degrees of freedom t test without much loss of accuracy. For ease of reference in the rest of this paper, we shall refer to this approximate degrees of freedom t simply as the Welch t test. In this paper, the lack of robustness of the Welch t test when the underlying distribution is nonnormal is demonstrated. A two-sample trimmed t statistic for unequal population variances is suggested and its performance is evaluated in comparison to the Welch t test under normality and under long-tailed distributions.

SUMMARY This paper considers a wide class of latent structure models. These models can serve as possible explanations of the observed relationships among a set of m manifest polytomous variables. The class of models considered here includes both models in which the parameters are identifiable and also models in which the parameters are not. For each of the models considered here, a relatively simple method is presented for calculating the maximum likeli- hood estimate of the frequencies in the m-way contingency table expected under the model, and for determining whether the parameters in the estimated model are identifiable. In addition, methods are presented for testing whether the model fits the observed data, and for replacing unidentifiable models that fit by identifiable models that fit. Some illus- trative applications to data are also included. This paper deals with the relationships among m polytomous variables, i.e. with the analysis of an m-way contingency table. These m variables are manifest variables in that, for each observed individual in a sample, his class with respect to each of the m variables is observed. We also consider here polytomous variables that are latent in that an individ- ual's class with respect to these variables is not observed. The classes of a latent variable will be called latent classes. Consider first a 4-way contingency table which cross-classifies a sample of n individuals with respect to four manifest polytomous variables A, B, C and D. If there is, say, some latent dichotomous variable X, so that each of the n individuals is in one of the two latent classes with respect to this variable, and within the tth latent class the manifest variables (A, B, C, D) are mutually independent, then this two-class latent structure would serve as a simple explanation of the observed relationships among the variables in the 4-way con- tingency table for the n individuals. There is a direct generalization when the latent variable has T classes. We shall present some relatively simple methods for determining whether the observed relationships among the variables in the m-way contingency table can be explained by a T-class structure, or by various modifications and extensions of this latent structure. To illustrate the methods we analyze Table 1, a 24 contingency table presented earlier by Stouffer & Toby (1951, 1962, 1963), which cross-classifies 216 respondents with respect to whether they tend towards universalistic values ( + ) or particularistic values (-) when confronted by each of four different situations of role conflict. The letters A, B, C and D in

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 the bivariate case, we shall write gn(tl, t2; P12) for gn(tl, t2; P) and Gn(tl, t2; P12) for Gn(t1, t2; p)If the random variables xj,..., xp follow the multivariate normal law with means zero, common variance o.2 and a correlation matrix (pij) and if (ns2)/o-2 is an independent x2 variable with n degrees of freedom, the random vector t = (tl, ..., tp), where ti = xi/s (i = 1, .. ., p) will have g,(t1,..., tp; P) as density function. Bechhofer, Dunnett & Sobel (1954) consider this distribution in connexion with a problem in the ranking of means of normal populations. Dunnett & Sobel (1954) give a formula for evaluating the probability integral when p = 2. Using this, they have prepared tables of the function Gn(h, h; ? 0.5) and its inverse. Dunnett & Sobel (1955) provide approximations for Gn(hl, ..., hp; P), valid when P satisfies certain conditions. In this paper we give an alternative formula for the evaluation of the probability integral. Though we too discuss only the bivariate case in detail, our method is of wider applicabilitv in the sense that it can be adopted to get the probability integral of the multivariate t-distribution of any dimension. We must mention here that the method of this paper is similar to that of Kendall (194 1) for evaluating the probability integral of the multivari'ate normal distribution. WVe have already mentioned that the multivariate t-distribution arises in the ranking of normal populations according to their means. We give more applications tow ards the end of this paper. It is shown how the multivariate t-distril)bition can be used in setting up simultaneous confidence bounds for the means of correlated( normal variables. Other applications are in constructing simultaneous confidence bounll(s for the parameters in a linear model and for future observations from a multivariate normnal distribution. Further applications will appear in a later paper.

Orthogonal polynomials with certain probability functions as weighting functions, have been employed by many authors in graduating sets of data. They have also been used in statistics in other contexts, e.g. by Shenton (1950, 1958) who has developedpowerful methods by their help to estimate parameters and to derive the efficiency of such estimated parameters by assuming that the maximum-likelihood functions can be expanded in a series of orthogonal polynomials. In this paper methods are developed of deriving the orthogonal polynomials of the discrete power series probability distribution which was defined by Noack (1950). These methods are due to van Heerden (1961), who has also derived the appropriate polynomials for the discrete factorial power series probability distribution, which will, however, not be treated in this paper.

are known to admit a sufficient estimator of 0 in realizations of the chain starting with a fixed state and consisting of a fixed number of transitions. This paper considers whether transition probabilities of the same form will admit a sufficient estimator of 0 in other finite regular, but not positively regular, Markov chains. For chains with an irreducible subset of two or more states, in which a realization starts from a fixed state and consists of a fixed number of transitions, these probabilities are found to admit a maximum-likelihood estimator of the function g(6) = -A'(O)/A'(0), which is sufficient and unbiased. There is some difference in chains with an absorbing state, in which realizations start from a fixed state but continue until the absorbing state is reached; in this sequential case, the maximum-likelihood estimator, with the number of transitions in the realization, together provide a sufficient estimator of the function g(O), which in general is no longer unbiased. We restrict ourselves to the particular case where a certain linear relation is satisfied. This gives rise to some simple stochastic matrices admitting a sufficient estimator of 0, which consist of probabilities of the forms 0 and 1 -0; in some of these cases, unbiased sufficient estimators of 0 reduce to known results of Girshick, Mosteller & Savage (1946). Some non-regular finite and infinite chains with absorbing states, associated with random walks, whose matrices consist of various patterns of probabilities 0 and 1 -0, are also found to admit a sufficient estimator of 0. The paper ends with the examination of such an example, arising in sequential estimation.

SUMMARY Recently Box & Cox (1964) and Fraser (1967) have proposed likelihood functions as a basis for choosing a transformation of data to yield a simple linear model. Here a simple, exact, test of significance is proposed for the consistency of the data with a postulated transformation within a given family. Confidence sets can be derived from this test. The power of the test may be estimated and used to predict the sharpness of the inferences to be derived from such an analysis. The methods are illustrated with examples from the paper by Box & Cox (1964). Box & Cox (1964) considered the choice of a transformation among a parametric family of data transformations to yield a simple, normal, linear model. They investigated two approaches to this problem and derived a likelihood function and a posterior distribution for the parameters of the transformation. Draper & Cox (1969) have found approximations for the precision of the maximum likelihood estimate. Fraser (1967) derived a different likelihood function which yields quite different inferences from those of Box & Cox (1964) in extreme cases where the number of parameters is close to the number of observations. Likelihood methods require repeated computations using a number of transforms of the original data. This can be troublesome if there is a multiparameter family of transformations. A further defect of the likelihood methods is that confidence limits and tests based on them have only asymptotic validity; the number of parameters must be small compared with the number of observations. This will not be the case for small data sets, paired comparison experiments and extreme cases. In the present paper, a method is proposed which has three possible advantages over direct calculation of likelihoods. Its main disadvantage is that it does not lead to such a clear graphical summary of conclusions as is given by a plot of a likelihood. The advantages are: (i) an 'exact' test of significance is obtained from which 'exact' confidence limits can be calculated; (ii) the amount of calculation is reduced if only one or a few transforms are to be tested; (iii) the precision with which the transformation can be estimated is capable of theoretical calculation.

It is the purpose of this paper to analyse a class of distribution functions that appears in a wide range of empirical data-particularly data describing sociological, biological and economic phenomena. Its appearance is so frequent, and the phenomena in which it appears so diverse, that one is led to the conjecture that if these phenomena have any property in common it can only be a similarity in the structure of the underlying probability mechanisms. The empirical distributions to which we shall refer specifically are: (A) distributions of words in prose samples by their frequency of occurrence, (B) distributions of scientists by number of papers published, (C) distributions of cities by population, (D) distributions of incomes by size, and (E) distributions of biological genera by number of species. No one supposes that there is any connexion between horse-kicks suffered by soldiers in the German army and blood cells on a microscope slide other than that the same urn scheme provides a satisfactory abstract model of both phenomena. It is in the same direction that we shall look for an explanation of the observed close similarities among the five classes of distributions listed above. The observed distributions have the following characteristics in common: (a) They are J-shaped, or at least highly skewed, with very long upper tails. The tails can generally be approximated closely by a function of the form

1.1. We are concerned in this paper with the problem of estimating the mean value ,u of a variable x in a population, by taking a sample which is in some way representative of the population. It has been realized since Bowley's paper (1926), and more particularly since Neyman's more comprehensive survey (1934), that a certain degree of precision in the estimate can often be obtained more economically by stratified random sampling (usually referred to merely as stratified sampling) than by unrestricted random sampling (usually called merely random sampling). In the stratified method, the population is divided into several strata, the sample size divided in some prearranged way among the strata, and sampling performed at random from each stratum. In unrestricted random sampling, a random selection is made from the whole population, and the method may be regarded as a particular case of stratification, where the number of strata is one. Some text-books deal briefly with stratified sampling. Wilks (1943) considers only infinite populations, and denotes by representative sampling what we should call a particular type of stratified sampling (see ? 1.2). The subject is treated by Kendall (1946, pp. 249-52), but he makes no comparison with unrestricted random sampling. We shall begin by introducing several well-known results which will be needed later.

We will be considering what Lindley (1968) in his paper on the choice of variables in multiple regression termed 'a prediction problem'. By this we mean that data from a regression experiment is analyzed in such a way that we can predict a future value of the dependent variable and choose which independent variables to use for the prediction. In this paper, the regression experiment is designed, i.e. the levels of the independent variables are selected. Our object is to find the optimal design. Although many papers have been published on obtaining optimal regression designs, the approach of this paper is different in that it uses Bayesian decision theory rather than orthodox (classical) methods. Lindley (1968) considered the prediction problem given the results of the regression experiment, and we shall be using some of his results in this paper.

Now that satisfactory and probably final solutions have been obtained for a wide variety of statistical problems concerned with a single normally distributed variable, more and more attention has recently been given to the solution of multivariate problems. The multiple correlation methods of the old large sample theory have been replaced in many instances by others for which " studentized " test criteria are available, often having sampling distributions that are already familiar in univariate problems. In a recent paper on " The statistical utilization of multiple measurements", R. A. Fisher (1938a) has shown the connexion between certain of these methods: the D2-statistic work of Mahalanobis, the discriminant function methods of the Galton Laboratory and the generalized " Student's " ratio of Hotelling. A similar very general problem was dealt with some time ago by S. S. Wilks (1932), while mention may also be made of two papers by D. G. Lawley (1938 a, b) and a paper by P. L. Hsu (1938). The purpose of the methods put forward is to obtain information regarding the mean values of a number, say q, of correlated variables in one or more, say k, populations from which random samples have been drawn. If we denote by x8 a value of the sth variable (s = 1, 2, ..., q), then in all this work it has been assumed not only that x8 is normally distributed, but that it has the same variance o2 in every population sampled. Further, it is assumed that if x. is a second variable the correlation coefficients psu between x8 and x. is the same in all populations. The estimates of variance and covariance required in order to "studentize" the function of the sample means are therefore obtained by pooling together the sums of squares and sums of products from all samples. While it is true that even if oa, and psu are not the same in all populations the error involved may not be very large, it is however important to have available some means of testing the basic hypothesis which assumes homogeneity throughout the populations. Such a test has been derived by S. S. Wilks (1932) by an extension of Neyman & Pearson's likelihood ratio method of approach. Hitherto the somewhat lengthy computations required to obtain the moments of the sampling distribution of the test criterion have probably discouraged its use. The objects of the present paper are as follows: (a) In the simple but commonly met case, where the k samples are of the same

The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

When estimating the size of a finite population, it is possible to consider, as an alternative to the capture-recapture method, a sequential scheme. Suppose an urn contains an unknown number, N, balls, initially all white. A single ball is drawn at random and if it is white it is painted black and returned to the urn, while if it is black, indicating that it has already been sampled at least once before, it is returned unchanged. Thus initially nearly all drawings will be of white balls, but after a long time mostly black balls will appear. Somewhere in between we wish to stop sampling and produce an estimate of N. This scheme was first proposed by Goodman (1949, 1953). The most recent treatment of this problem is by Samuel (1968, 1969), who considered asymptotic distributions of sample sizes and maximum likelihood estimates for each of four stopping rules. The crucial question of the choice between these rules remained unanswered, however, since the balance between cost of sampling and risk due to inaccurate estimation was not considered. It is precisely this which is the aim of the present paper. A bibliography of previous, non-Bayesian, work is given by Samuel (1968). Since preparing the first draft of this paper, the author has seen an unpublished report by D. G. Hoel and W. E. Lever in which this problem is formulated in a Bayesian framework, and a small numerical solution given for a quadratic loss function, and with an upper limit on the value of N.

Professor William Hovanitz called my attention to the following problem: An observer catches butterflies, marks them, and immediately releases them. It is assumed that a butterfly, no matter how many times it has been caught before, has the same susceptibility to capture as any other butterfly in the population which is supposed stable while the captures are being made. Records are kept of f, the frequency of cases in which the same butterfly is caught x times, x = 1, 2, ..., until a total of s captures of r different butterflies have been made. (Efx = r; Zxfx = s.) The number, fo, of butterflies which escape is not observed; the problem is to estimate from the values offx the total population n of butterflies on the area assumed well defined. The estimation of biological populations by means of capture-recapture data is by no means a new problem, though papers dealing with it from the mathematical-statistical point of view are largely quite recent. (In particular, see the papers by Leslie & Chitty (1951), Bailey (1951), Moran (1951, 1952), and the bibliographies quoted by them.) However, the experimental conditions and the mathematical models for the present study arpear to differ in essential ways from those previously considered. The important point of departure is that each butterfly on being netted is immediately marked (with a spot of nail polish) and released. The butterflies (Colias eurytheme) were caught in one of two isolated alfalfa fields, which they inhabit, in southern California. Each catch was made during the same day at times when the butterflies were freely flying. Thus, it seemed reasonable to assume that the population was stable during a catch. The experimenter, Prof. Hovanitz, endeavoured to give each butterfly an equal chance of capture, walking in straight lines across the field and deviating in direction before reaching a boundary only when he noticed that a butterfly just caught tended to fly down his path. One check of the suitability of a mathematical model is to test the agreement of the experimental results with respect to the number of butterflies caught once, twice, etc., with those predicted from the model. I will return to this point at the end of the paper. Two mathematical models seem appropriate to serve as a basis for discussion of this estimation problem. It is of some interest to see that both lead to approximately the same estimates with little difference in their precision for large samples. It may be of more interest that for both models in which the population size is regarded as a parameter, though maximum likelihood estimates exist and agree substantially with moment estimates in all sixteen of the actual field experiments for which I have data, nevertheless with increasing sample size meaningful solutions of the likelihood equation do not exist.

SUMMARY This paper discusses from a Bayesian viewpoint some aspects of the estimation of latent roots and vectors of the covariance matrix of the bivariate normal distribution. The joint distribution of (i) the angle of the canonical transformation, and (ii) the ratio of the larger root to the total variance is considered in detail and illustrated by an example. Also discussed is the problem of making inferences about the larger roots, and several simple approximations to the distribution are considered. Finally, a generalization is given of one of the approximation methods to latent roots of higher dimensional covariance matrices. In this paper some aspects of the estimation of latent roots and vectors of the covariance matrix of the bivariate normal distribution are considered from a Bayesian viewpoint. In ? 2 the joint posterior distribution is obtained for (i) the angle 0 associated with the canonical transformation, and (ii) the ratio V of the larger root to the total variance, i.e. the sum of the two roots. Some properties of the joint distribution and the associated marginal distributions are discussed and a numerical example is given. Attention is then turned, in ? 3, to the problem of making inferences about the larger root and several simple approximations to its distribution are considered. In particular, it is demonstrated that the distribution can be closely approximated by a scaled inverted x2 distribution upon equating moments.

In a recent note Bartlett (1954) has summarized various approximate x2 tests (see also other papers referred to therein). Our object in this paper is to extend certain of these, listed as III a, b, c, which may be regarded as tests involving the latent roots of sample covariance and correlation matrices. We shall be interested in those cases where the effects of the k largest latent roots have been removed and where a hypothesis of equality of the remaining roots is made. Such hypotheses are made in principal components and factor analyses.

The purpose of this paper is to show the geometrical significance of the moments of order statistics derived from normal populations. It appears that these, as well as the moment-generating function of the square of any order statistic, are intimately related to the contents of the members of a class of hyperspherical simplices. Further geometrical interpretations, together with the extension of the present results to bivariate moments and other properties, will be provided in subsequent publications. The problem of order statistics in normal populations has been extensively considered in the literature. For example,t Tippett (1925) gives the second, third and fourth moments of the extreme order statistics for a few sample sizes. Hojo (1931) examines the sampling variation of the median, quartiles and interquartile distance in samples from normal populations and computes a large number of integrals for this purpose. Cole (1951) produces a very simple recurrence relationship between the 'normalized' moments which enables the normalized moments for all samples of size no greater than n to be obtained, by successive differencing, from the 'normalized' moments of the extreme order statistics in samples of size m < n.4 Hastings, Mosteller, Tukey & Winsor (1947) give, among other results, the means, variances, covariances and correlations of order statistics in samples of ten or less from a normal population, some of the results being given to only two decimal places because of the extreme labour of the computation. Jones (1948) and Godwin (1949a,b) have obtained exact values for some of the lower moments. In particular, Godwin (1949a, p. 283) proceeds in a more systematic manner, and approaches most closely the essential idea upon which this paper is based. In general, it may be said that many statisticians have attacked the problem in a rather disjointed and fragmentary manner, usually from the small sample end, but have failed to develop a systematic attack which shall at the same time throw light on the interconnexion between the moments and enable the computation of the moments to become an economic proposition. It is believed that these conditions are met by this paper.

Let Xl, X2, X3 and X4 obey a quadrivariate niormal distribution with zero means, aind let P4 be the value of the quadrivariate normal integral, i.e. the probability that Xl, X2, X3 and X4 are simultaneously positive. The generalized tetrachoric series for P4, as given by Aitken (unpublished), Kendall (1941, 1945) (see also Kendall & Stuart, 1958, pp. 350-4), and Moran (1948), are not well suited for computation. For the case in which all six correlation coefficients are equal, the series has been sumlmed approximately by McFadden (1956). For special numerical values of the correlation matrix, exact results for P4 have been given by Schlafli (1858, 1860), Anis & Lloyd (1953), and Plackett (1954). Methods for numerical integration in more general cases have been given by Plackett (1954), Ihm (1959), and John (1959). Numerical methods for integration when all the correlation coefficients are equal have been provided by Ruben (1954) and Moran (1956). In this paper we present two series expansions for P4 which are well suited for computation. The first case occurs when X1, X2, X3 and X4 are successive measurements from a stationary Gaussian Markov process (with zero mean), or, equivalently, when the inverse of the correlation matrix has zero elements except on the main diagonal and immediately adjacent to it. The second case occurs when the correlation matrix itself has zero elements except on the main diagonal and adjacent to it. We shall then show that these two cases are related by a simple transformation. In a previous paper, McFadden (1955, eq. (39)) has expressed P4 in terms of the various product moments of the set of random variables Yi which assume the sign of the normal variables Xi. Let Y.=1 when X. O' ---1 when Xi< Oj (i= 1,2,3,4). (1)

SUMMARY The point of change in mean in a sequence of normal random variables can be estimated from a cumulative sum test scheme. The asymptotic distribution of this estimate and associated test statistics are derived and numerical results given. The relation to likelihood inference is emphasized. Asymptotic results are compared with empirical sequential results, and some practical implications are discussed. The cumulative sum scheme for detecting distributional change in a sequence of random variables is a well-known technique in quality control, dating from the paper of Page (1954) to the recent expository account by van Dobben de Bruyn (1968). Throughout the literature on cumulative sum schemes the emphasis is placed on tests of departure from initial conditions. The purpose of this paper is to examine a secondary aspect: estimation of the index T in a sequence {xt}, where the departure from initial conditions has taken place. The work is closely related to an earlier paper by Hinkley (1970), in which maximum likelihood estimation and inference were discussed. We consider specifically sequences of normal random variables x1, ..., xT, say, where initially the mean 00 and the variance o2 are known. A cumulative sum, cusum, scheme is used to detect possible change in mean from 00, and for simplicity suppose that it is a one-sided scheme for detecting decrease in mean. Then the procedure is to compute the cumulative sums t

SUMMARY This paper is concerned with the application of several closed sequential procedures to the testing of an exponential parameter. In particular, the procedures of Armitage (1957), Schneiderman & Armitage (1962a,b) and Anderson (1960) are considered. In certain of these cases either the properties or bounds on the properties of the sequential test can be obtained without approximations. There are certain situations in which the risk of a long sequential experiment is unacceptable. Some common examples are to be found in the area of medical trials and in various types of industrial experimentation. Such situations have brought about the study of alternatives to the sequential probability ratio test (SPRT) in which a procedure's boundaries are closed. One class of such tests is the 'restricted' procedures of Armitage (1957) and the 'wedge' plans of Schneiderman & Armitage (1962a, b). Another test which also belongs to this category is a modification of the SPRT as described by Anderson (1960) and by Donnelly in his unpublished Ph.D. thesis. These tests, which are concerned with normal means, require an approximation of the random walk by a diffusion process for an evaluation of their properties. In this paper we shall apply the above type regions to the testing of the parameter from an exponential distribution and show that in certain cases the properties of the test can be obtained without approximations.

IN genetical practice we are constantly presented with large numbers of small samples from populations consisting of several well-defined classes. For example in the mouse we can readily obtain hundreds of litters containing anything from one up to about twelve members. Their totals may agree satisfactorily with expectation on a Mendelian basis, for example J coloured, i white, or 9 grey, H black, i white. But we desire to know whether the individual litters can be regarded as random samples from such a population. In addition the problem of homogeneity may arise. That is to say the population as a whole may not conform to any particular expectation. But we may desire to know whether the litters can be regarded as random samples of the population given by the totals. It has long been known that when the numbers expected in any observation are small, the distribution of x2 departs from that given by Pearson (1900). The mean appears sometimes, but not always, to be equal to the number of degrees of freedom. But the variance is no longer exactly equal to twice that number. Exact expressions for it in certain cases have been given by Pearson (1932) and Cochran (1936). These are based on an ingenious application of the theory of multiple contingency by Pearson. It will be shown in this paper that the first few moments can often be calculated by entirely elementary methods involving nothing more advanced than the multinomial theorem. In an accompanying paper (Griineberg and Haldane, 1937) they will be applied to actual data on mice. We first study the distribution of x2 in a n-fold table with n 1 degrees of freedom, then in a (m x n)-fold table with m (n-i) degrees of freedom. For genetical work we are particularly interested in the (n x 2)-fold table with n degrees of freedom. As a limiting case of the 2-fold table with 1 degree of freedom we derive the moments of the variance of samples from a Poisson series, and thence the distribution of x2 in a n-fold table with n degrees of freedom. The important case of the (m x n)-fold table with (m 1) (n 1) degrees of freedom remains to be investigated. Consider a sample of 8 individuals falling into n classes. Let the expected and observed numbers in these classes be

On account of the unique property of samples from a normal population that the ratio n+1 (x-Fb)/V(n + 1)/s (where jt is the population mean, x = x1/(n + 1) and ns2 =Z(Xi= 1 is the ratio of a normal deviate to a stochastically independent estimate of its variance, Student's t-test is a suitable test of significance for the mean of a normal population. However, in a variety of cases, it is necessary to test for the mean of a population which does not follow the Gaussian law. Efforts have, therefore, been made to see how far Student's distribution may be used for the purpose in non-normal populations. Due, mainly, to the analytical difficulties of the problem, no extensive theoretical discussion has yet been given. Thus, Pearson & Adyanthaya (1929), Rietz (1939) and Nair (1941) have given experimental treatments, while the theoretical discussions of some others (Rider, 1929; Perlo, 1933; Laderman, 1939) have dealt only with trivially small sample sizes. The papers by Bartlett (1935) and Geary (1936, 1947) give results true for any sample size, though they are based on certain assumptions and approximations. The present paper deals with the population considered by Geary in his 1936 paper, subject to the same approximations. The second contribution by Geary (in which is derived the t-distribution in samples from a population which departs more from normality than that considered in the 1936 paper) came to my notice too late to be made use of in the present work; but it is proposed to consider it later on. Geary (1936) has obtained the distribution of the ratio (x -It) (n + 1)/s in the case of an asymmetrical population, whose fourth and higher cumulants are zero, by neglecting squares and higher powers of the third cumulant. We know from this how far the probability of an error of the first kind (i.e. the probability of rejecting the null hypothesis when it is true) in such a population differs from that for a normal distribution, provided we may neglect the square of the standardized third cumulant yl. Here again, on account of analytical difficulties, it is not possible, except for very small sample sizes, to consider the effect of terms containing higher powers of yl. However, we can assume the result derived by Geary to be correct for very small values of yl, as also for large sample sizes-but in such cases the deviation from values of the normal theory is practically negligible. Even then, it is of interest to know whether, in using the usual tables of the t-test (based on the normal distribution), we are committing the greater error in the probability of an error of the first kind or in that of an error of the second kind. In the present paper are derived the values of the probability of an error of the second kind (and hence, of the power of the test) when the usual t-tables are used to define the critical region. It may be mentioned here that this problem is only a special case of a general investigation, on which the writer is engaged, into the effect, on statistical tests, of differences between the actual and the assumed distribution laws of the universe sampled. The solution of these problems is hampered by analytical difficulties in the derivation of the probability laws (and particularly of power functions), and the present case is one of the few in which a mathematical, though only approximate, solution has been found possible.

SUMMARY The probability integral for the distribution of range of a sample from a population whose distribution can be represented by the first few terms of an Edgeworth series has been obtained in this paper. The numerical values of the corrective functions arising due to nonnormality are tabulated. The new theoretical results are compared with the earlier results, where available. The distribution of range of samples from nonnormal populations was first studied empirically by Pearson & Adyanthdya (1928) and later, among others, by Pearson (1950), Cox (1954) and David (1954). These studies have been limited mainly to the mean range and to the probability integral in some simple nonnormal cases, and from these likely effects of nonnormality on the distribution of range have been conjectured. Singh (1967) obtained some theoretical results regarding the expectation and the variance of range of samples from a population whose distribution can be represented by the first few terms of an Edgeworth series. These results provided some additional information regarding the effects of parental excess and skewness on the mean and variance of the range. In the present paper the probability integral for the distribution of range of samples from the same type of population has been obtained and evaluated for small samples to examine the effects of parental excess and skewness.

(1) A PAPER* by Eleanor Pairmian anld Karl Pearson, F.R.S., was published in 1918 on "Corrections for the Moment-Coefficients of Limited Range Frequency Distributions when there are Finite or Iinfinite Ordinates and any Slopes at the Terminals of the Range." In the second part of that paper, dealing with cases of asymptotic frequency, the auxiliarv curve selected to give the first five frequencies was