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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.

SUMMARY The processes discussed in this paper are models for populations of independent identical individuals reproducing by fission, with the alternative possibility of death. The probabilities for birth and death are considered to be age-dependent, so that under suitable assumptions the populations have a probability to become extinct, which is neither zero nor one. It is the purpose of the present paper to study conditional probabilities for such populations where the condition is that they do ultimately become extinct. This is done by working in terms of a space of family trees, and the results include conditional probabilities for the life-lengths of individuals and the number of their offspring. An example is given, and an application is made to a model of bacterial toxicity. For a Markov process possessing a set of transient states, and one or more sets of absorbing states, there is a simple method for deriving probabilities conditional on absorption in a given set, from unconditional probabilities for the same process (Breny, 1962; Kemeny & Snell, 1960, p. 64; Waugh, 1958). This method takes a particularly simple form for the Markovian binary-fission-or-death branching process when the condition is that of ultimate extinction, and it has been applied in this form to a model of the toxic effect of a bacterial invasion (Puri, 1966). In the present note, a related method is developed for nonMarkovian age-dependent processes. The results take rather simple forms when expressed in terms of the distributions, conditional and otherwise, of the life-lengths of individuals and of the number of their children. In the process discussed in this note the life-length distributions may take general forms, whereas in the Markovian case they are all negativeexponential. An example for Erlangian life-lengths is given, and the method is used to extend results of Jagers (1967) for the integral of the population size. Incidentally, Kendall (1966) has shown how to obtain conditional probabilities for Markovian birth processes that tend to infinity, where conditioning is on the random variable W = lim {Zt1f'(Z1)}, Zt being the population size at time t. The variable W is discussed

This paper examines the structure of some early examples of serial factorial changeover designs given by Quenouille (1953). Quenouille listed two designs for 3 treatments, 6 periods, 18 subjects and three designs for 5 treatments, 8 periods, 16 subjects. The present paper shows that Quenouille's cyclic method of construction can be extended to designs for any number t of treatments, 2t periods, and t2 subjects. The enumeration of these designs is shown to be equivalent to the enumeration of Eulerian trails in certain directed graphs.

SUMMARY The paper describes the analysis of variance of some experimental designs in which there may be several levels of splitting of plots, and in which the treatments at any level may be non-orthogonal to blocks. The construction of the designs, based on the Kronecker product method as used by Kurkjian & Zelen (1962), is discussed first. Then the method of analysis, based on the ideas developed by Nelder (1965a, b), is outlined. The paper shows that some potentially useful designs, not previously discussed in any detail, can be analysed quite simply using these ideas. The combination of information from several sources, and the derivation of unbiased estimates of error, are also mentioned.

This paper deals with some applications of a general theory for the analysis of factorial experiments as reported by the authors in the June 1962 issue of the Annals of Mathematical Statistics. General expressions are given for the usual quantities associated with the analysis of variance for the cases where simple treatments or factorial treatment-combinations are applied to Randomized Blocks, Balanced Incomplete Blocks, Group Divisible designs, and a wide class of Kronecker Product designs. The main point of the new theory is that, for a wide class of the more practical designs, the complete analysis can be carried out almost by inspection of the normal equations, with no requirement for inverting the normal equations. The complete version of this paper is published in BIOMETRIKA, Vol. 50, Parts 1 and 2, June 1963.

Association between two variables has in the past been measured or tested by several coefficients, among them (i) Product moment correlation coefficient. (ii) Spearman's rank correlation coefficient. (iii) Kendall's alternative rank coefficient T. (iv) Probability of concordance. The efficiency of each of these coefficients in detecting association between two variables is not in general known, but it is known that none is uniformly better than the others. Conjectures have been made that coefficients (ii) and (iii) behave in much the same way for reasonable distributions, but as pointed out by Daniels (1950) a sample ranking can be considered for which (ii) and (iii) have very different values. If the population is such that rankings of this type have appreciable probability, it would be expected that tests based on (ii) and (iii) would have different efficiencies. It would obviously be desirable if the performance of these coefficients could be studied for a fairly general class of distributions, in order to gain insight into the relationship between the coefficients. The class of bivariate distributions obtained from monotonic transformations of variates originally possessing a bivariate Normal distribution has been studied by Fieller, Hartley & Pearson (1957) by means of random samples of correlated Normal deviates. They concluded that for sample sizes 30 and 50 with correlation greater than 0-6 in the original bivariate normal distribution Kendall's coefficient was probably better than Spearman's. Correspondence with one of the authors, E. S. Pearson, indicates that the Fisher-Yates statistic mentioned in their paper is better than either rank correlation coefficient within this same field. Another such general class of distributions has been considered by Konijn (1958), namely, the class derived from linear combinations of two independent variables. It is the purpose of this paper first to propose a general class of bivariate distribution functions; secondly, to study how the various coefficients of association compare in efficiency with each other and with a maximum likelihood estimator of the parameter of association; and finally to use the general results to determine which are the best coefficients to use in a number of special cases. The efficiency measure used in this paper will be the asymptotic relative efficiency, in the neighbourhood of independence, as defined by Pitman (1948). The form of the bivariate distribution function proposed in this paper is an extension of an idea of Morgenstern; see, for example, Gumbel (1958). Morgenstern (1956) proposed F(x) G(y) [1+ c{1 F(x)} {I G(y)}] as a bivariate distribution function having F(x), G(y) as marginal distribution functions, and Gumbel noted that this differed from the bivariate normal distribution if F(x) =_ (x) and G(y) --4(y) where (D is the normal error function.

SUMMARY If we have k binomial samples of different sizes, we may sometimes be interested in the question of homogeneity, i.e. we may want to know whether the k samples all came from binomial distributions with the same parameter p. A similar question of homogeneity may arise if we have k samples from multinomial or Poisson distributions. This paper, which is the first of a series of two, treats the binomial and multinomial situations; the second paper will treat the Poisson case. Homogeneity tests already exist for the problems just mentioned, but these existing tests apparently were not constructed with any optimal power properties explicitly in mind. These papers approach the problem of homogeneity testing by attempting to construct tests having maximal power against certain reasonable alternative hypotheses. Some new tests result from this approach; these tests will be described and numerical illustrations will be presented. The traditional tests (i.e. the usual x2 tests) will also be discussed. For the binomial problem, all tests which are considered are applicable to the situation (frequently arising in genetics) in which some or all of the k sample sizes are small numbers (even as small as 2 or 3). Section 1 of this paper is concerned with testing for homogeneity for the binomial case; a number of biological applications are presented. In ? 3, we show briefly how the new test which is introduced in ? 1 may be generalized to the multinomial case. All of the more technical details have been relegated to the Mathematical Appendices, which form the last part of the paper.

In recent years, many papers have appeared describing Monte Carlo techniques which have been applied to structured cluster problems (bond and site) on various twoand threedimensional lattices; see, for example, Frisch, Sonnenblick, Vyssotsky & Hammersley (1961), Frisch, Hammersley & Welsh (1962), Dean (1963). Estimates have been obtained for the expected cluster size, the probability that a point belongs to an infinite cluster, and the critical probability at which infinite clusters occur. In this paper we discuss a twodimensional unstructured cluster problem, i.e. one in which there is no underlying lattice structure as there is in the structured case. It is interesting to note the similarity between the structured and unstructured cases. Consider an infinite plane on which are scattered discs of radius R whose centres are points distributed according to the Poisson law with density A, i.e. the probability that the centre of a disc lies in a small area &A equals A &A + o(6A). A cluster of size n is defined as a set of n discs each of which overlaps at least one other member of the set and none of which overlaps a disc which is not a member of the set. Although the probability of a cluster of size n and the expected number of discs in a cluster depend on the two variables A and R, dimensional analysis suggests that they are functions of only one composite variable t = AR2. This can also be verified by evaluating the probability of a cluster of size n for small values of n. Let pn(t) be the probability of a non-zero cluster of size n, and let e(t) be the expected number of discs in a non-zero cluster. There exists a critical value t = tc at which e(t) becomes infinite (Gilbert, 1961). For t > t, infinite clusters occur. We obtain a graph of e(t) as a function of t and estimate t, by extrapolation. As in the structured cases the difficulties of a direct approach prove insurmountable and hence a Monte Carlo solution is used. Gilbert (1961) has considered this problem in relation to communication networks and the spread of contagious disease. He uses the notation E = 4lTt. He is interested in P,'(E), the probability that a particle belongs to an infinite cluster. This is clearly zero for E < E,. By approximating P,,(E) by PN(E) which is the probability that a particle belongs to a cluster of size N or more, and evaluating PN(E) by a Monte Carlo procedure, he obtains an estimate Ec = 3X2, te = 0255. Section 2 describes a Monte Carlo technique to evaluate e(t) and a graph of e(t) as a

In an earlier paper (Rao, 1959), the author discussed the method of least squares when the observations are dependent and the dispersion matrix is unknown but an independent estimate is available. The unknown dispersion matrix was, however, considered as an arbitrary positive definite matrix. In the present paper we shall consider a class of problems where the dispersion matrix has a known structure and discuss the appropriate statistical methods. More specifically the structure of the dispersion matrix results from considering the parameters in the well-known Gauss-Markoff linear model as random variables. Let Y be a vector random variable with the structure

SUMMARY An extension of the negative binomial model as a species frequency distribution is given by allowing the shape parameter k to take values between -1 and 0. Different estimation methods are assessed for this extended negative binomial model and the logarithmic series model. Tables for standard errors are given. Many successful attempts have been made to fit mathematical models to populations of many species. Fisher's logarithmic series model (Fisher, Corbet & Williams, 1943), Preston's log normal model (Preston, 1948), the negative binomial model (Brian, 1953) and McArthur's broken stick model (McArthur, 1957) have all proved useful, giving a reasonably good fit to biological data. The purpose of this paper is to give an extension of the negative binomial model by allowing the parameter, usually symbolized by k, to take values between - 1 and 0, and to assess different estimation methods for this extended negative binomial model and the logarithmic series model. The extended model seems to fit well in situations where the negative binomial fails and has usually been substituted by the log normal model, which is more complicated to handle mathematically.

SUMMARY This paper presents a numerical method of statistical inference which overcomes some mathematical difficulties encountered in the nonstationary Poisson process by taking full advantage of modern computing equipment. The maximum likelihood estimator of an exponential polynomial rate function has moments equal to the corresponding sums of powers of the observed event times. A numerical determination of this function is demonstrated. The information matrix, a simple function of the moments of the rate function, can also be estimated by the sums of powers. Finally, a goodness-of-fit test is derived from the relation between sums of powers of event times and moments of the rate function. Computer programs which perform all the necessary calculations have been prepared and are available from the author.

(1920), Fisher (1941), Haldane (1941), Anscoinbe (1950) and Bliss & Fisher (1953), and is extensively used for the description of data too heterogeneous to be fitted by a Poisson distribution. Observed samples, however, may be truncated, in the sense that the number of individuals falling into the zero class cannot be determined. For example, if chromosome breaks in irradiated tissue can occur only in those cells which are at a particular stage of the mitotic cycle at the time of irradiation, a cell can be demonstrated to have been at that stage only if breaks actually occur. Thus in the distribution of breaks per cell, cells not susceptible to breakage are indistinguishable from susceptible cells in which no breaks occur. Methods for estimation of the parameters of the truncated distribution are considered in this paper. The corresponding problem of estimation of the truncated Poisson distribution has been discussed by David & Johns-on (1952), who also discuss the present problem.

It is often possible, in certain types of mass production, to use a large sample of articles for simple routine inspection and to find with ease the articles with more extreme values of the characteristic measured. Examples of this are articles which undergo a routine check on their length or weight, in which case extreme values can be sorted out either by sight, or by use of GO-NO GO checks on a balance set at two suitable weights. In these cases, a great deal of labour can be saved, if the dispersion is estimated from these extreme values, which may comprise only about 5 % of the total. Such an estimate of dispersion may be used in controlling variability by specifying limits for this estimate. One miiethod of specifying the variability, which avoids the complication of subdividing the sample, is to lay down limits for the difference between the sum of the r highest and r lowest values observed in the sample. In this paper it will be shown how the mean, variance, and also higher moments of this difference can be found. Approximate formulae, which are reasonably easy to calculate, are given for the mean and variance of the difference. These should be satisfactory for most p)ractical purposes. In Table 1 are given exact values of the mean and variance of the difference in the case when the parent population is Gaussian (normal) for selected sample sizes and values of r. The mean and variance with other parent populations may be calculated by applying equations (22) and (25) to Tables 3 and 4.

Fisher (1928) expressed the distribution function of the non-central %2 in terms of Bessel functions with imaginary argument. Wishart (1932) and Tang (1938) evaluated the probability integrals of the non-central %2 and F distributions; they involve a heavy amount of labour. In this paper Laguerre series expansions of these distributions are derived. In ? 2 series expansions of the distributionp (X) of X = ,y'2 (denoting a non-central chi-squared by x'2) is developed in terms of Laguerre polynomials, namely

SUMMARY Consider a finite set of points, located on the circumference of a circle. Several tests have been proposed of the hypothesis that the points constitute a random sample from a uniform distribution. In this paper we study a test statistic defined as the maximal number of points that can be covered by some semicircle. Exact and asymptotic distributions under the null hypothesis, and under a certain alternative hypothesis, are given together with some tables. A related test statistic is studied briefly. An expression is obtained concerning most powerful invariant tests of the hypothesis of a uniform circular distribution. In 1965, Dr G. Borenius described an unpublished experiment with a bubble chamber, where points representing events were observed through a circular window. A natural hypothesis was that the events occurred at random with a constant probability density within the circle. In one case it was observed that 67 out of 100 events fell within a suitably chosen semicircle. The question then arose whether this asymmetry should be judged inconsistent with the hypothesis of a uniform distribution. More generally, suppose that n points are observed and that each point is moved radially to the circumferences of the circle. We then have a sample of n points on the circumference and want to test the hypothesis that the underlying probability distribution is uniform over the circumference. The test statistic suggested by the foregoing paragraph is the maximal number of points in the sample that can be covered by a suitably chosen semicircle. In the following this test statistic will be denoted by N. We reject the hypothesis if N is too large. Many other tests of the same hypothesis have been proposed. For example, the classical Kolmogorov-Smirnov test has been adapted to circular distributions by Kuiper; see Kuiper (1960) and Stephens (1965). Watson (1961) did the same thing for the Crame6r-von Mises test. A detailed study of the null distribution of Watson's test statistic has been made by Stephens (1963, 1964). For a general review of statistical methods in connexion with circular distributions, see Batschelet (1965). The problem of determining the distribution of N under the hypothesis is purely combinatorial. It was solved by Borenius for sample sizes n up to n = 15 and the general solution was inductively conjectured by him. He thus found that the above-mentioned observation, N = 67 for a sample of size n = 100, corresponds to a level of significance P = 1U6 0/. Dr S. Johansen, University of Copenhagen, has told the author that he, too, has found the null-distribution of N. This was done in connexion with an application to the study of

SUMMARY This paper is devoted to the early history of the law of large numbers. An outline of the prehistory of this law is given in ? 1. The algebraic part of J. Bernoulli's theorem is presented in a logarithmic form and the lesser known role of N. Bernoulli is described in ? 2. Comments on the derivation of the De Moivre-Laplace limit theorems by De Moivre, in particular, on the inductive character of his work, on the priority of De Moivre as to the continuous uniform distribution, on the unaccomplished possibility of Simpson having arrived at the normal distribution and on the role of Laplace are presentedin ? 3. The historical role of J. Bernoulli's form of the law of large numbers is discussed in ? 4.

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)

1. SUMMARY A number of statistical procedures involve the comparison of a 'regression' mean square with a 'residual' mean square using the normal-theory F distribution for reference. The use of the procedure for the analysis of actual data implies that the distribution of the meansquare ratio is insensitive to moderate non-normality. Many investigators, in particular Pearson (1931), Geary (1947), Gayen (1950), have considered the sensitivity of this distribution to parent non-normality for important special cases and a very general investigation was carried out by David & Johnson (1951a, b). The principal object of this paper is to demonstrate the overriding influence which the numerical values of the regression variables have in deciding sensitivity to non-normality and to demonstrate the essential nature of this dependency. We first obtain a simple approximation to the distribution of the regression F statistic in the non-normal case. This shows that it is 'the extent of non-normality' in the regression variables (the x's), which determines sensitivity to non-normality in the observations (the y's). Our results are illustrated for certain familiar special cases. In particular the well-known robustness of the analysis of variance test to compare means of equal-sized groups and the notorious lack of robustness of the test to compare two estimates of variance from independent samples are discussed in this context. We finally show that it is possible to choose the regression variables so that, to the order of approximation we employ, non-normality in the y's is without effect on the distribution of the test statistic. Our results demonstrate the effect which the choice of experimental design has in deciding robustness to non-normality.

SUMMARY This paper provides numerical and analytical solutions to the problem of estimating the rate of a Poisson process. Optimal designs are obtained for various loss functions and the method of analysis is valid for any other loss function. The cost of sampling plays a fundamental role and since there are many practical situations where there is a time cost and an event cost, a sampling cost per observed event and a cost per unit time are both included.

In this paper exact and approximate methods are given for computing the distribution of quadratic forms in normal variables. In statistical applications the interest centres in general, for a quadratic form Q and a given value x, around the probability P{Q > x}. Methods of computation have previously been given e.g. by Box (1954), Gurland (1955) and by Grad & Solomon (1955). None of these methods is very easily applicable except, when it can be used, the finite series of Box. Furthermore, all the methods are valid only for quadratic forms in central variables. Situations occur where quadratic forms in non-central variables must be considered as well. Let x = (x1, ..., xx)' be a column random vector which follows a multidimensional normal law with mean vector 0 and covariance matrix E. Let s = (,t, . . ., ,,7)' be a constant vector, and consider the quadratic form Q = (x + ,)' A(x + ,u). If E is non-singular, one can by means of a non-singular linear transformation (Scheff6 (1959), p. 418) express Q in the form rn 2 Q =E ArXhr; (1 r=1

SUMMARY The following general problem is considered: fit the inverse cumulative distribution function F-1(y) by a polynomial expansion in terms of a more tractable function G-1(y). It is shown that the computation of the coefficients of this expansion need not depend upon the evaluation of F-1(y) for specific values of y, but instead can be based on the evaluation of the cumulative F(x). If C-1(y) is chosen to be - log (1 - y), the solution in this particular case is shown to be based upon the Laguerre polynomials. Applications of the above methods are briefly described for such problems, as: random number generation, order statistic moment and product moment calculation, as well as the smoothing of the sample cumulative. There are many theoretical and practical applications of the inverse cumulative distribution function F-1(y). For this reason several investigations have been conducted in search of either exact and tractable expressions for F-1(y) or else, in cases where tractable exact expressions do not exist, for suitable approximations. In this paper the general problem of finding 'suitable' approximations to the inverse cumulative will be considered. In ? 2 the general solution is given for the problem of best approximation of F-1(y) by a polynomial expansion in powers of a second function G-1(y). When G-1(y) is chosen as a more tractable expression than F-1(y), the resulting expansion can be used as a convenient approximation of F-1(y). It is also shown that the computation of the coefficients of this expansion does not rely upon the evaluation of F-1(y) for various specific values of y, which would be the case if ordinary least squares fit were attempted. In ?? 3, 4, and 5 examples are given of the use of the general result of ? 2 to find approximations to the inverse Gaussian cumulative over various subintervals of the unit interval. An increase in accuracy over previously published results is demonstrated. Although there are many applications of the inverse cumulative, three are of particularly current interest. These are: first, the use of approximations of F-1(y) for order statistic moment and product moment evaluation; second, the approximation of specific inverse cumulatives for use in random number generation; and third, the smoothing of the sample cumulative. Tractable expressions for F-1(y) have been used to calculate the moments and product moments of the order statistics which in turn have been used to find Best Linear Unbiased, i.e. B.L.U., estimators. The most extensive research on this topic, that conducted by David & Johnson (1954) was based upon the use of a truncated power series approximation to F-1(y). As an alternate approach, Tarter (1965), has given approximate expressions for

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

SUMMARY The Durbin-Watson test on autocorrelation is based on the least squares residual vector. It is well known that the distribution of this vector depends upon the regression matrix wlhich implies that tabulation of the test statistic's significance points is senseless. The best linear unbiased scalar test circumvents this difficulty and gives a test statistic whose distribution does not depend upon the regression matrix; it is an exact test. However, some objections can be made against this test. In this paper, a new exact procedure which meets these objections is presented. A method to compute the test statistic is outlined. Powers of the new procedure for some examples are computed and compared with the corresponding powers of the Durbin-Watson test and the best linear unbiased scalar test.

The idea of basing tests on the sample distribution function is a natural one. The Kolmogorov-Smirnov tests are of this nature. Blum, Kiefer & Rosenblatt (1961) made use of this approach to construct distribution free tests of independence. In this paper this method is further applied to the common two-sample problems of location and dispersion, the k-sample problem of location, and to the problem of dependence in bivariate samples. The rationale of the method is the following. Write the departure from the null hypothesis (against which the test must be sensitive) in terms of the true distribution functions and regard the mean value of this departure as the parameter of interest. The sample estimate of this parameter, expressed in terms of sample distribution functions, is then proposed as test statistic. This statistic is generally only dependent on the ranks of the samples and is consequently distribution free. Specifically this approach leads to Wilcoxon's two-sample test, a k-sample extension of Wilcoxon's test which is slightly different from the Kruskal-Wallis (1952) extension, the test of Ansari & Bradley (1960) for differences in dispersion, which is also a special case of a procedure proposed by Barton & David (1958), and to a test of dependence in bivariate samples which comes out to be a linear function of the rank correlation coefficients of Spearman and Kendall. The alternative k-sample extension of Wilcoxon's test has the same asymptotic relative efficiency properties as the Kruskal-Wallis test; it is however consistent against a slightly wider class than the latter. As is to be expected, the alternative test of independence behaves for large samples in much the same way as the tests of Spearman and Kendall.

One of the practical difficulties in estimation by maximum likelihood arises from the fact that, except in special cases, the resulting equations are complicated and not easily solved. The present paper approaches this problem by deriving an expansion based on moment approximations to the likelihood equations. These are also not easy to solve explicitly, but they furnish a method of deriving the efficiency of the method of moments in terms of an expansion of the determinant of the information matrix. The efficiency of moment fitting as applied to Pearson's distributions has been discussed by Fisher (1921), who more recently treated the case of the negative binomial (Fisher, 1941).

This paper is concerned with the estimation of the covariance parameter of the bivariate Poisson distribution. It is shown that the method of moments has low efficiency for distributions with appreciable correlation, and an iterative method of solving the likelihood equation is described. A further method of estimation is described which is more efficient than the method of moments for certain regions of the parameter space. Finally, the numerical results of applying the maximum-likelihood method to two sets of data are given as illustrations.

Hotelling's suggestion of a 'circular' definition for the serial correlation coefficient was followed by considerable progress in the distribution theory of such modified statistics by R. L. Anderson (1942), Koopmans (1942), Dixon (1944), Madow (1945) and others. The exact distribution is known for the circular coefficient of any lag from an uncorrelated normal process and, more generally, from a circularly modified normal process of autoregressive type. Quenouille (1949) obtained by the same method the exact joint distribution of circular coefficients of different lags. The exact distributions are complicated, and a simple and accurate approximation to the distribution of the circular coefficient with known mean was found by Dixon (1944) and Rubiin (1945) for the uncorrelated normal process. It was extended to the case of a circular Markov process by Leipnik (1947) following a method due to Madow (1945). The approximation depends on the device of smoothing summation over a discrete set of roots by an approximating integration. Quenouille (1949) conjectured a similar approximate form for the joint distribution, but Watson (1951) and Jenkins (1954) showed that the conjectured form could not be correct. Jenkins developed the correct analogous approximation for the joinlt distribution of coefficients of lags 1 and 2 with known means. Without circular modifications the distributional theory is difficult and the field is largely unexplored. For testing independence T. W. Anderson (1948) gave an approximate table of significance points for non-circular lag 1 coefficients with known and fitted means. Watson & Durbin (1951) introduced modified non-circular definitions of the coefficients which have R. L. Anderson's distribution in the uncorrelated case. The case of an unmodified autoregressive process has not been much discussed, though a method due to Bartlett (1953, 1954) is available for obtaining approximate confidence intervals for the parameters, and Quenouille's (1947) approximate goodness of fit tests should be noted. In the present paper an approach based on the method of steepest descents is adopted to derive the known approximate distributions and to generalize them.* (For an account of the method see, for example, Jeffreys & Jeffreys (1956), Daniels (1954).) The analogue of Leipnik's approximation is found for the distribution of an unmodified coefficient of lag 1, both with known and fitted mean, when the process is of unmodified Markov type. The approximate joint distribution of m successive partial serial correlations is found for an autoregressive process of the mth order, circular modifications being used. The work on the unmodified Markov process could be extended to the general case but we have not done this.

SUMMARY An analytic expression is derived for the average probability density function of the time interval between two consecutive events in a continuum of time, for the case in which the instantaneous rate of occurrence of events contains a term varying sinusoidally with the time in addition to a constant term. Expressions are obtained also for the average values of the mean and variance of the time interval between two consecutive events. An application of the theory to the analysis of phenomena for which the data consist solely of the times of occurrence of isolated events in a continuum of time is discussed briefly. The mathematical statistics presented in this paper originated in a study of cosmic ray extensive air showers. In particular, a search was made for the possible existence of a periodic variation in the experimentally observed occurrence rate of extensive air shower events at a fixed position on the earth's surface. Of special importance is the case in which the average rate of occurrence of events is roughly comparable with the anticipated frequency of the periodic modulation. The present theoretical investigation indicates how the amplitude and frequency of a periodic modulation can be estimated solely from a knowledge of the measured time intervals between successive events. Since the theory may be applied to the analysis of other phenomena, for which the data consist only of the times of occurrence of isolated events in a continuum of time, the arguments are presented without specific reference to the cosmic ray problem. The first objective of the theory is to determine the probability density function (P.D.F.) of the time interval between two consecutive events in a continuum of time, for the case in which the instantaneous rate of occurrence of events contains a term varying sinusoidally with the time in addition to a constant term. In the special case in which the instantaneous occurrence rate is constant, the P.D.F. of the time interval between two consecutive events may be derived at once from the usual form of the homogeneous Poisson distribution. The particular instantaneous occurrence rate considered here generates a non-homogeneous Poisson distribution, from which the average value of the P.D.F. of the time interval between two consecutive events may be derived. This average P.D.F. is then used to find the average values of the mean and variance of the time interval between two consecutive events. Finally, consideration is given to an application of the above theory to the statistical analysis of any phenomenon which manifests itself as the occurrence of discrete events in a continuum of time, and for which a periodic modulation of the occurrence rate is anticipated. The measured time intervals between successive events are used to calculate the observed average probability of obtaining a time interval between two consecutive events which lies

SUMMARY Neyman (1959) has defined a locally asymptotically most powerful test, called optimal asymptotic test, of composite statistical hypotheses. This test is particularly applicable when either the standard methods of testing are not available or when they are too difficult to compute. However, Neyman (1959) restricted his studies to the case of testing a onedimensional parameter. In this paper his results are generalized to the case of testing a k-dimensional parameter. This necessitates introducing the notion of a locally asymptotically most stringent test (L.A.M.S.T.). The theory is developed in the spirit of Neyman's work, using his ideas and definitions omitting a number of details. It is illustrated by testing a general linear hypothesis involving multinomial parameters. We also give the L.A.M.S.T. when the observations are not independent identically distributed. An example from bio-assay is also given.

SUMMARY This paper explores analytically the outer portion of the Bayes continuation region in sequential sampling problems for the Bernoulli process with a beta distribution on the parameter p and for the Poisson process with a scaled gamma distribution on the parameter A. We assume that there are two actions, that the cost per Bernoulli trial or per unit time of Poisson observation is constant, and that utility is suitably linear. Our concern is with continuation points which cannot be reached from any continuation point corresponding to a smaller number of Bernoulli successes or Poisson occurrences. These points form an inaccessible region which need not be considered in calculating the rest of the continuation region. In length this region is approximately 50-63 % of the whole continuation region when the sampling cost is small, 63 % in the Poisson case. We also give computable nearly tight bounds on the inner end of the region, which may be used to start backward induction to obtain the rest of the continuation region. These results follow from more detailed results concerning the outer portion of continuation region. 1. OUTLINE The Bernoulli process is discussed first. In ? 2, we state in detail the problem considered and the results obtained for this process. The proofs are given in ? 3. The Poisson process is discussed in ? 4. ? 5 contains remarks on various matters including further improvement of the bounds and different kinds of inaccessible regions. A very few directly relevant references are given in ?? 2 and 4, but well known facts about the general framework are stated without reference to the literature.