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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 More recently Goodman, in an unpublished paper, has derived several further modifications of his estimator.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 To define a likelihood we have to specify the form of distribution of the observations, but to define a quasi-likelihood function we need only specify a relation between the mean and variance of the observations and the quasi-likelihood can then be used for estimation. For a one-parameter exponential family the log likelihood is the same as the quasi-likelihood and it follows that assuming a one-paramet'er exponential family is the weakest sort of distributional assumption t'hat can be made. The Gauss-Newton method for calculating nonlinear least squares estimates generalizes easily to deal wit'h maximum quasi-likelihood estimates, and a rearrangement of this produces a generalization of the method described by Nelder & Wedderburn (1972). This paper is mainly concerned wit'h fitting regression models, linear or nonlinear, in which the variance of each observation is specified to be either equal to, or proportional to, some function of its expectat'ion. If the form of distribution of the observations were specified, t'he method of maximum likelihood would give estimates of t'he parameters in t'he model. For instance, if it is specified that the observations have normally distributed errors with constant variance, then the method of least squares provides expressions for the variances and covariances of the estimates, exact for linear models and approximate for nonlinear ones, and these estimates and the expressions for their errors remain valid even if the observations are not normally distributed but merely have a fixed variance; thus, with linear models and a given error variance, the variance of least squares estimat'es is not affected by the distribution of the errors, and the same holds approximately for nonlinear ones. A more general situation is considered in this paper, namely the situation when there is a given relation between the variance and mean of the observations, possibly with an unknown constant of proportionality. A similar problem was considered from a Bayesian viewpoint by Hartigan (1969).We define a quasi-likelihood function, which can be used for estimation in the same way as a likelihood function. With constant variance this again leads to least squares estimation. When other mean-variance relationships are specified, the quasilikelihood sometimes turns out to be a recognizable likelihood function; for instance, for a constant coefficient of variation the quasi-likelihood function is the same as the likelihood obtained by treating the observations as if they had a gamma distribution.

The estimation of parameters in an absorbing Markov chain has been discussed by Gani (1956), Bhat & Gani (1960), Bhat (1961) and possibly by other authors. Asymptotic theory for the distribution of maximum-likelihood estimators applies if a large number of independent replicates of the chain is available. These replicates, however, could be considered as occurring sequentially in time, so that the chain has implicitly been altered in such a way that the absorbing state has been replaced by an 'instantaneous return' state; once it is reached, a new replicate is commenced starting from the original initial state. In this paper, we concern ourselves with the asymptotic theory of maximum-likelihood estimators from a single realization of truly absorbing chains. It is clear that if the number of states is kept fixed there can be no asymptotic theory, since with probability one, an absorbing state will be reached after a finite number of transitions, and no further informa. tion can be obtained by continuing observation. In ? 2, we show by means of two simple examples that asymptotic theory may, at least in some cases, be available if the number of non-absorbing states is large. In the remaining sections, forming the main part of the paper, we discuss a more complex example, a population genetic model of Moran (1958a). We do not prove that the conjectured asymptotic theory holds for the estimation of the parameter, but produce numerical evidence from simulation studies to support this conjecture. Further research is needed to clarify the general problem of inference in absorbing Markov chains. The problem can be made to depend on the theory of positively regular chains but the latter is itself incomplete for this purpose.

Recent papers by Hotelling & Pabst (1936), Pitman (1937), Kendall (1938) and Kendall, Kendall & Babington-Smith (1939) discuss the distribution of the correlation coefficient when members of the sample corresponding to the two variates are permuted randomly relative to one another. In the case of rank correlation, the characteristics of the population sampled are generally unknown, and a significance test has to be based on the distribution obtained from the sample in this way. Hotelling & Pabst prove that as the sample size is increased, Spearman's p tends to follow a normal distribution law. Kendall's measure of rank correlation, r, in which all possible corresponding pairs in two given rankings are assigned marks + 1 according to whether they agree or differ in order, follows a specially simple distribution law which tends rapidly to the normal form and becomes highly correlated with Spearman's p for samples of moderate size. The present paper discusses the properties of the class of correlation coefficients I obtained on replacing Kendall's marks + 1 by a more general system of scores. By an empirical argument Kendall et al. showed it to be likely that the correlation between r and p is

Systematic sampling methods have been used in a variety of fields, notably in forestry and ecological studies, because of their convenience in the field and their advantages over a randomized scheme if the survey is also to be used for mapping. Their use was justified by work such as that of Hasel (1938) and Osborne (1942), who examined the sampling errors of random, stratified random and systematic surveys, by subjecting detailed forestry survey data to analysis by these three methods. These analyses showed that the systematic designs were generally the most efficient. They left unsolved the problem of estimating the error of the systematic sample, although Osborne's paper gave a method based on computing the mean square serial correlation which gave reasonable agreement with experimental results. More recently, Finney (1948), also using data from forest surveys, has compared the variances of systematic samples (computed by the overlapping group method to obtain reasonable precision), random samples and stratified random samples with one and two members per stratum. In these cases Finney showed that the variance of the systematic sample differed little from the variance of a stratified random sample of the same size with one observation per stratum, but was appreciably smaller than the variance obtained with half the number of strata and two observations per stratum, which appeared to be the most efficient system giving an unbiased estimate of the variance without supplementary information. Further investigation of the variance of systematic sampling with various models seems likely to be useful. Yates (1948) examined the sampling variance for a number of particular cases and gave a form (p. 362) which, with supplementary information, provided an estimate of the sampling variance of the means of a systematic sample, or, alternatively, subject to certain assumptions, provided an upper limit to the error. Jowett (1952) gave a method for the determination of the variance of a systematic sample, on the assumption that the observations were derived from a stationary process. This was derived from earlier work by Cochran (1946) and Quenouille (1949) in the same field. In this paper we shall develop a method similar to Jowett's, but involving rather weaker assumptions; we shall also derive a form for the variance of samples with spacings less than or equal to that of the observed data (which is not covered by Jowett's method); this includes the case where the variance is derived from a single sample. This does not conflict with the often-stated principle that a systematic sample cannot by itself provide an estimate of error, since the assumptions which are made about the population are equivalent to supplementary information. Many alternative assumptions could have been made, and the justification of the particular ones chosen must be that they appear likely to apply to a wide variety of data and in the cases considered below yield results which agree with

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

are possible. We set out in this paper five different methods of randomization in a circle and two possible criteria which may be used to test for randomness. A method of obtaining the moments of these criteria is given and an approximation to the P.D.F. is suggested. Certain practical difficulties have been encountered in applying the theory in that there is uncertainty in the identification of the chromosomes, the shadow of the wall of the nucleus is not always visible and when visible is only approximately circular and so on. We give here theory only and reserve the practical applications for another paper 2. NOTATIONS The plane area is assumed to be a circle of unit radius and an arbitrary axis of reference is supposed. n random chords are supposed drawn on this circle. If we consider a random point (R, 0) (O < R < 1, 0 < 0 < 27T), we write its P.D.F. asf(R, 0). If we consider a random chord it is sometimes convenient to consider the perpendicular, p, to it from the centre, O < p < 1, and the angle a the perpendicular makes with the reference line. The P.D.F. of the centre of the chord we will denote by g(p, a). It is sometimes convenient to work withf (R, 0)

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.

In the field of biology the problem of estimating population parameters such as population size, death rate, birth rate, etc., for animal populations is obviously an important one. Various capture-tag-recapture models have been developed to estimate these population parameters with a minimum number of assumptions on the underlying population. One such method, the multiple-recapture census, has been the topic of many papers and is described briefly as follows. The experimenter takes a sequence of random samples a,, a2, ... , a8, say. The members of each sample a* are tagged and returned to the population before taking the next sample. Thus the members of a2, a3, ..., a, can be classified according to when, if at all, they have been captured before. Although several models have been developed from different basic assumptions, three papers in particular by Darroch (1958, 1959) and Jolly (1965) give the most general treatment of this method in the form of exact, fully stochastic models which lend themselves readily to the method of maximum-likelihood estimation. The first paper of Darroch's deals with the closed population, i.e. a population in which there is neither augmentation due to immigration (or birth) nor departure due to death (or emigration). This population is usually dealt with separately as the algebra involved is quite different from that which arises in more general populations. The main assumption made, which in fact underlies all capturerecapture models, is that marked and unmarked individuals have the same probability of being caught and as far as the author knows no test statistic has been given for testing this assumption. In ? 4 of this paper we shall consider this problem and a likelihood-ratio test statistic will be derived which gives an approximate test of the above assumption. In Darroch's second paper he derives, for a population in which there is either immigration or death (but not both), modified maximum-likelihood estimates of the population parameters that are almost unbiased and asymptotically efficient, that is efficient for a certain class of reasonable estimates. For the population in which there is both immigration and death he gives only the probability generating function for the model and derives moment estimates of the parameters. However, Jolly (1965), tackling this problem from a different viewpoint, gives a very elegant solution to the problem of finding maximumlikelihood estimates of the unknown population parameters and gives the means and variances of these estimates. In this paper we shall consider this general population with both immigration and death and set up a model which differs slightly from that of Darroch and Jolly in that certain parameters are treated as unknown constants rather than as random variables. The notation and approach to the model is essentially that of Darroch's while the method of solution is similar to that given in Seber (1962). The estimates obtained from this model will be compared briefly with those obtained by Jolly.

This paper is concerned with the average delay to vehicles on the minor road at a junction where the traffic on the major road has absolute priority. It is assumed that vehicles on the major road arrive at random, and pass through at intervals of not less than a time f1. Vehicles on the minor road arrive at random and pass through at intervals of not less than f2. Further, they cannot enter the junction within a time ax after the previous vehicle on the major road. Thus one minor-road vehicle can pass through a gap of duration between aX and a + /32 in the major-road traffic, two can pass in a gap of between a + fl2 and a + 2/2, and so on. A formula is derived for the average delay to vehicles on the minor road when the system is in statistical equilibrium. Extensive tabulations of this formula have been made, and an extract from these is given. The practical implications of the results are not discussed in the paper, but it is hoped to compare the delays with those obtained with traffic-light control. The problem is similar to those of queueing with priority customers, of the 'service repeat' variety, but there is one important difference: at road junctions of the type considered there is a considerable loss of service time when changing from one type of ' customer' to the other. Typical values of /,J and /12' the two service times, are of the order of 1 or 2 sec., but a gap a of about 7 sec. between two priority 'customers' is required for any non-priority 'customers' to be served. An analysis of a similar problem has been given by Winsten (Beckmann, McGuire & Winsten, 1955), but this was based on more restrictive assumptions. Some special cases of the result obtained here were given by Tanner (1953).

and A ij is the cofactor of a,j in the determinant A I = j a1 ijj At a later date Wilks (1932) defined a generalized variance and found the appropriate A-criteria for testing certain hypotheses concerning the means, variances, and covariances of k normal multivariate populations from which k independent samples have been drawn. These criteria were developed more fully by Pearson and Wilks in a subsequent paper (1933) for the case of two variates, but though the sampling distributions obtained were in certain cases relatively simple the arithmetical calculations required for practical application were, in general, not of a very simple nature.* In this paper we shall find a quantity which may be regarded as a generalization of Fisher's z and which provides a test suitable for dealing with certain generalized analysis of variance problems, having the advantage of being easy to apply.

SUMMARY The paper deals with the estimation of the parameters of a variance component model for two-way crossed classifications with replication. Estimates are needed for row, column and interaction components of variance, and also for the ratios of these variances to the error component of variance. A balanced design with k(k > 2) observations for all subclasses may sometimes be considered extravagant for the purpose. Alternative designs in which duplicates are taken from a selection of subclasses and a single observation is taken from each of the rest, are proposed in the paper. A simple method of analysing these schemes is suggested, and some numerical results on the efficiency of these designs are presented.

In the analysis of variance for a one-way classification it is customary to test the hypothesis that the samples have come from populations with the same mean. The object of the present paper is to discuss the best way of making this test when the unknown population means are subject to order restrictions. A general theory of tests of homogeneity for meanls under ordered alternatives has been developed in three earlier papers (Bartholomew, 1959 a, b, 1961): these are referred to in this paper as I, II and III, respectively. However, apart from a brief discussion in I, this earlier work relates to the case where the population standard deviations are known a priori. This is a natural assumption to make when investigating the theoretical properties of tests for means but is unrealistic in many practical applications. In particular, in the analysis of variance the standard deviations are not usually known, although they can be estimated from the data. In ? 2 the X2-test, first introduced in I and extended in III, will be generalized to cover this case also. The tests based on scores discussed in III may also be adapted for use in the analysis of variance. It, will be shown that a close link exists between the latter and the distribution-free test proposed by Jonckheere (1954). A number of factors must be taken into account when choosing a test of which power is one of the most important. Asymptotic results have been obtained for all the tests mentioned above and they are used in ? 3 to make power comparisons. It is instructive to examine more closely the kind of practical situation which gives rise to ordered alternatives in the analysis of variance. The possibility of ranking the class meanis implies that they correspond to different levels of one or more underlying variates. To take a simple example, suppose that there are k classes with means /,l, /L21 .. ., 1at with the structu re

1-1. General background. In a basic paper on multivariate tests of linear hypotheses, Wilks (1932) applied the method of the likelihood ratio to the problem of testing the equality of mean vectors in k multivariate populations. The general approach of this early paper has been extended and expanded in the literature until, now, a procedure is available for testing multivariate linear hypotheses in a completely general form (see, for example, Anderson, 1958). For the problem of computing the power of these tests, however, no such general procedure exists. This is due to the fact that the noncentral distribution of the test criterion has not been expressed in a numerically feasible form. A first step towards the derivation of the non-central distribution, however, was taken when Anderson & Girshick (1944) attacked the problem of finding the distribution of the Wishart matrix in the non-central case. The result led Anderson (1946) to the derivation of the moments of the criterion for testing the hypothesis of the general multivariate regression problem (or rather the equivalent Wilks-Lawley hypothesis). Anderson's results were valid for a certain class of alternatives, which he called linear and planar. The moments for the linear alternative involve an infinite sum of expressions containing gamma functions and for the planar alternative, involve a triple infinite sum of the same type of expressions. This appears to be the extent of the progress on the problem of obtaining the power of the multivariate likelihood-ratio test of the general linear hypothesis.t

Tests for the presence of outlying observations generally have one of the following aims: (a) to screen data in routine fashion preparatory to analysis (the problem of 'rejection of outliers'); (b) to sound an alarm that outliers are present, thus indicating the need for closer study of the data-generating process; (c) to pin-point observations which may be of special interest just because they are extreme. The proper treatment of outliers has long been a subject for study. Numerous teststatistics have been devised, mostly from intuitive considerations, and their percentage points tabulated, on the assumption of a common normal parent population. A historical survey is unnecessary here especially as a number of review papers have recently appeared (Dixon, 1962; Ferguson, 1961a; see also Anscombe, 1960). In spite of the long literature on the subject much more needs to be known about the performance of the various statistics in use for the non-null situation when outliers are present. Case (a) will not be treated in this paper but has received some attention in the references just cited. In ? 2 we list and discuss several measures of performance, including the power function, which are relevant to cases (b) and (c). These measures are illustrated on four well-known statistics expressible as ratios of the extreme deviate from the sample mean to the population standard deviation cr or estimates thereof. It is assumed that the underlying variation is normal with a single true outlier present which differs in mean from the remainder of the sample. Some of the statistics we consider have been studied under these assumptions by experimental sampling (Dixon, 1950, 1962; Ferguson, 1961 b). A theoretical attempt, unfortunately erroneous, at evaluating the power of the extreme deviate test (cknown) was made by David (1956). A table there given is corrected and extended in ? 3. The simple measure P2 of ? 2 is used in ? 4 for the construction of charts comparing all four statistics. Of particular interest is the comparison of Nair's (1948) (externally) studentized extreme deviate test with the corresponding test in which the estimation of cr is based on (internal) information from the sample at hand as well as on the independent (external) information needed for Nair's statistic.

A brief review, with references, of the literature on capture-recapture theory is given in Jolly (1963). More recently, Cormack (1964) gives a solution, including asymptotic variances, for a specific situation involving the marking and release of a non-random sample of fulmar petrels. His model is stochastic and will be referred to in ? 4. Seber (1962, 1965) has also produced some interesting solutions, and these together with Darroch (1958, 1959) are most directly relevant to our present problem. Darroch (1958, 1959) shows that in a fully stochastic model with either immigration (often called dilution) or death (or emigration), the population parameters can be easily estimated by maximum likelihood. For the more general case when death and immigration are operating simultaneously he derives estimation equations by equating certain observations to their expectations, but does not give variances or covariances for the estimates. In a later paper, (Darroch, 1961), he considers estimation for a closed population consisting of different strata. Seber (1962) establishes a stochastic model for what he calls the multi-sample single recapture census in which an individual cannot be recaptured more than once. This situation arises, for example, when the recaptures are made in the course of hunting or fishing. He allows for both death and immigration in the population, provides explicit maximumlikelihood estimates of the parameters with variances, and suggests tests for certain of the assumptions. In a second paper, (Seber, 1965), he considers a multiple-recapture model differing only slightly from that of Darroch (1959), with both death and immigration. Again, he provides explicit maximum-likelihood estimates of the parameters with variances. A test is also given for equi-catchability in a closed population of individuals with different capture histories. The first purpose of the present paper is to derive a general probability distribution designed to fit the majority of capture-recapture problems involving a 'single' population. The word 'single' here denotes a population covering an area within whose boundaries the animals (or, in general, individuals or members) are free to move and to mix with others of their kind, but which is regarded as a single area in respect of which parameters are to be estimated. The type of situation which is thus excluded by this definition is one where the population is split into a number of defined areas, and separate population estimates are required for each area as well as for numbers of animals moving from one area to another. The single population, however, need not be homogeneous but may consist of different classes of animals behaving in different ways. The other assumptions underlying the model are stated with the notation in ? 2, and the generalized probability distribution is derived in ?3.

SUMMARY Various expectations concerning the convex hull of N independently and identically distributed random points in the plane or in space are evaluated. Integral expressions are given for the expected area, expected perimeter, expected probability content and expected number of sides. These integrals are shown to be particularly simple when the underlying distribution is normal or uniform over a disk or sphere. In two recent papers Renyi & Sulanki (1963, 1964) have given expressions for the expected area, perimeter, and number of vertices of the convex hull of N independently and identically selected random points in the plane. In these papers, limit theorems for asymptotically large N receive the greatest attention. Here the emphasis will be on the development of convenient formulae for fixed values of N. Some new results for random convex hulls in the plane are derived, such as the expected probability content, and also various expectations concerned with random convex hulls in three and higher dimensions. Special attention is given to the case of normally distributed points and also to the case of points drawn uniformly from an ellipse or an ellipsoid. Historically, calculating the expected probability content of three random points in two dimensions is known as 'Sylvester's problem', and has been solved explicitly for many different distributions by Deltheil (1926, p. 42). The corresponding problem of four points in three dimensions is connected with the name of Hostinsky (1925). A discussion of Sylvester's problem is given in Kendall & Moran's monograph, Geometrical Probability (1963). 2. THE EXPECTED NUMBER OF VERTICES, FACES, AND EDGES

THE present paper deals with experiments on the control of insects in the field. In such experimental work the problem to be investigated is whether more insects survive on plots which have been subjected to one treatment than on plots subjected to another. It will be shown in the present paper that the numbers of insects found per plot must vary in such a way that one cannot, strictly, subject the results to the analysis of variance, and it is proposed to find how the data may be transformed so that analysis of variance becomes applicable. Such transformation has been discussed by Bartlett (1936a,b) in connexion with entomological experiments, and by Tippett (1934) in connexion with industrial experiments.

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.

The most familiar method of testing the hypothesis that an observed spatial distribution of points in the Euclidean plane is a realization of a Poisson point process, or in practical terminology that the points are distributed 'completely at random' is based on the Index of Dispersion. In recent years, however, statistical ecologists have shown interest in what are known as distance, or nearest neighbour methods for studying the pattern of plant distributions. These involve choosing a set of n sampling points in a fixed sampling area B according to some rule, and measuring the distances Xij from the ith sample point to the jth nearest plant to it, measurements to plants outside B being permitted if necessary. The observations Xij are then used in estimators of the density of the spatial point process of which the plant distribution is a realization, or in testing hypotheses about its form. Distance methods have been discussed by Skellam (1952), Strand (1953), Moore (1954), Hopkins (1954), Thompson (1956), Matern (1959), Pielou (1959), Mountford (1961), Kendall & Moran (1963, p. 38), Keuls, Over & De Wit (1963), Persson (1964) and Holgate (1964, 1965 a, b). Tests of randomness based wholly or partly on distance methods have been proposed by Skellam, Moore, Hopkins and Pielou (see also Mountford's paper). These tests are discussed in the present paper, with particular emphasis on their power to detect certain kinds of departure from randomness. In Holgate (1965a) two more tests are proposed and discussed.

ciated R(x) was devised in terms of 0(I /x), 0 (2/x), ... for x > 0, which is much superior to the classical expansion (1.1) from the point of view of accuracy. In a still more recent paper (Ruben,t 1963), a further new expansion for R(x), which is not only asymptotic but also convergent when x > 0, was devised in terms of rational fractions. The aim of the present paper is to devise a new expansion for R(x) in terms of irrational fractions. The latter expansion, like the preceding expansion, is asymptotic and convergent for all x > 0. It should be noted that a different class of irrational fraction approximations, based on the convergents of Laplace's continued fraction representation of R(x) (x > 0), has been given by Shenton (1954). For high or moderate positive values of x, the current expansion converges very rapidly to R(x), and numerical examination appears to suggest that the convergence is then a good deal more rapid than the Shenton sequence of approximations. Moreover, the former expansion has the advantages that the individual terms are easily determined explicitly in algebraic form and upper bounds for the truncation errors are readily established. These bounds are, however, conservative. We may refer here also to the paper by Ray & Pitman (1963) in which the authors approximate Mills' ratio over the two separate ranges [0, 1], [1, oo] of the argument. Some interesting new asymptotic expansions (not in terms of rational or irrational fractions) are also obtained for the ratio.

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

The method of frequency-moments was developed primarily with the object of overcoming certain difficulties when fitting growth curves to observed data. In the author's original paper (1947) examples were given on how to fit exponential, logistic and Gompertz functions. In the second part of the same paper it was pointed out that the method could also be applied to the graduation of frequency-distributions and examples of normal, lognormal and Pearson Type VII curves were presented. Yule's (1938) investigation, based on an early suggestion of Karl Pearson, has come to the author's notice recently. The fitting process described by Yule is essentially the same as my method of frequency-moments. The derivation of the standard errors given on the following pages is based partly on Yule's work. In this paper we are more interested in the problem of estimating the parameters of a given type of frequency-distribution by the method of frequency-moments than in fitting the proposed distribution to a set of observations, though this aspect is dealt with in the last section. The present investigation has been limited to the case of frequency-distributions only, and the section comparing the maximum likelihood solution with the frequencymoment method has been confined to a Pearson Type VII population.

The problem of biserial correlation arises when one is sampling from a bivariate normal population in which one of the variables has beeii dichotomized, giving rise to only two observable values, say 0 and 1, and one wishes to use this dichotomized sample to estimate, or to test hypotheses concerning, the correlation coefficient p of the original bivariate normal distribution. The problem of biserial correlation occurs frequently in psychological work, especially in test construction and validation. The term biserial correlation was introduced by Karl Pearson (1909), who was the first to perceive the statistical importance of this particular type of problem. He proposed as an estimator the sample biserial correlation coefficient. The asymptotic variance of this estimator was derived by Soper (1913). Much literature exists on the subject of how best to compute Pearson's coefficient. In this connexion the reader should see Du Bois (1942), Dunlap (1936) and Royer (1941). Prof. Harold Hotelling realized some years ago that the existing methods for dealing with the problem of biserial correlation were far from satisfactory, and suggested to the author that the whole situation be reconsidered. The results of this examination are contained in the present paper. ? 2 contains a list of most of the notation which has been adopted, and ? 3 deals with the mathematical model. In ? 4 the question of maximum likelihood is treated. Asymptotic variances are derived for the estimators (^O and p. The asymptotic variance for , is compared with the approximate expression arrived at by Maritz (1953) when he considered a somewhat restricted model. Both expressions are shown to achieve their minimum value at ()O = 0 when p is fixed. Matters concerning asymptotic normality and asymptotic efficiency are also considered. An appraisal of r*, the sample biserial correlation coefficient, is given in detail in ? 5. It is shown to have asymptotic efficiency for estimating p which is 1 when p = 0but which approaches 0 when I p I approaches 1. The well-known fact that r* may be greater than I is pointed out and some notion of the magnitude of r* is obtained by a consideration of the product-moment correlation coefficient r. Asymptotic normality of r* is verified by the use of a theorem of Cramer. The asymptotic standard deviation is tabulated at the end of the paper (Table 2). A proof is given for the customarily assumed fact that the asymptotic variance has a minimum for fixed p when w = 0. For the case O = 0 an approximate variance stabilizing transformation is derived. Calculations pertaining to this transformation may

In a previous paper (Fucks, 1952, cited as I), several possibilities were outlined to describe the style of samples of literary texts by means of mathematical characteristics. The computing of practical examples, however, was confined to the most simple ones of these style characteristics. Most of them were derived from the frequency distributions of the properties of elementary units of the text themselves. It was, however, pointed out briefly, that the properties of groups of elementary units of the text can also be used to define characteristic numbers and, as an example, the values of an lik-matrix were given. To explain the aim of this paper, we assume the text to be divided into elements. The elements will be denoted by the numbers 1, 2, ..., A. It is perfectly arbitrary what parts of the text are chosen as elements: syllables, words, parts of the phrases, whole phrases, chapters, etc., or substantives, adjectives, verbs, etc., or cases of substantives, tenses of verbs, etc., or metric elements, and so on. The elements are to be considered as characterized by different marks, one or several of them for each element. These marks (or properties) can be chosen, within certain limits, in an arbitrary manner. However, to clarify our conception with the help of a special example, we shall generally in this paper, as in paper I, choose the words of a text as elements and the numbers of the syllables of the words as the interesting properties of these elements. Consequently, we shall now assume that the text is broken down into the words; and that these have been mixed together and then picked up, one after the other, perfectly at random and given the numbers 1, 2, ..., A. Theoretically, if we should repeat this procedure often enough, we should obtain the natural order of the original text as one of the different arrangements arising from picking up the words at random. We conclude, therefore, that we do not restrict the generality of our conclusions, if we start by marking the words of the text in their original order by the numbers 1, 2, ..., A. Now, let us assume that we have computed the values of pi, the proportion of the A words of a given text having i syllables and also the various characteristic numbers calculated in I, namely, i, the mean number of syllables per word; S =- E pilogpi, the 'entropy'; and i the 'trace', s. Considering these results, we ask how much of the structure of the text is described in an exact manner by these characteristic numbers. The answer is that our characteristic numbers do not tell us much more of the text than the vapour of a mixture of several fluids tells us about the fluid-structures proper, i.e. the real structure of the text does not enter into the characteristic numbers so far studied in detail. What we have taken into account with the characteristic numbers actually computed in I is the choice of a certain number, z, of words out of the reservoir V of the words of the language in which the text is written. The words enter into our characteristics as formed grammatically. But the grammatical forming process is contained in our characteristic numbers only in a rudimentary

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

The problem of estimating directly the spectral density function f(A) (or g(A)), previously considered by Daniell (see the discussion to the paper by Bartlett, 1946), Bartlett (1950) and Grenander (1951), will be discussed further in this paper. Grenander & Rosenblatt (1952, 1954) have recently investigated also the problem of constructing an entire confidence band for the integrated density function g(A); for further suggestions on this problem, which will not be considered here, see Bartlett (1954).

SUMMARY The effect of non-normality and the effect of treatment slope interaction in the analysis of covariance is studied. Unlike the analysis of variance F-test the analysis of covariance F-test is found to be appreciably affected by non-normality even in balanced classifications. The degree of sensitivity to non-normality is determined by the distribution of the concomitant variables. The violation of the assumption of no treatment-slope interaction does not seem to have serious effect on the F-test in large samples, but a much more serious effect appears to be that of a quadratic component of regression on the F-test. Atiqullah (1962) has shown that the extent to which the analysis of variance F-test is influenced by non-normality depends on the structure of the design matrix, A, in the usual linear model for the problem. In balanced set-ups the design matrix, A, possesses the property of quadratic balance which is characterized by the matrix A (A'A) 1-A' having its diagonal terms all equal. This property practically nullifies the effect of non-normality on inferences drawn from the F-test in the standard orthogonal cross-classified set-ups, such as balanced two or more-way arrangements, Latin squares, balanced incomplete block designs, etc. But in the analysis of covariance the linear model contains, besides the design matrix A, a matrix of concomitant observations and therefore the sensitivity of F-test to non-normality will be determined both by the structure of the design matrix A and the nature of the concomitant matrix X. In balanced classifications, it is thus the concomitant variables alone which will determine sensitivity to non-normality. This fact also follows from the discussion of Box & Watson (1962) who have discussed robustness to non-normality of regression tests and have shown that it is the extent of non-normality in the independent variables which determines sensitivity to non-normality. In the first part of the present paper we investigate in more detail the influence which the concomitant variables have in determining sensitivity to non-normalityin the analysis of covariance F-test and obtain expressions for judging the effect of non-normality in practical situations. In the second part of the paper we discuss how far the analysis of covariance, as usually applied on the presumption of no interaction between treatments and slope, will be sensitive to situations entailing interactions or to a model containing quadratic gradient.

An important characteristic of highly infectious diseases is that they tend to produce groups of cases occurring within households rather than separate cases scattered throughout the community. If all the cases in a household were due to some single source of infection such as typhoid-infected water, then we should expect the total number of cases to follow a binomial distribution. On the other hand, if the disease were introduced into the household by one of its members and then transmitted from person to person, a very different situation would arise. For a disease involving a short period of high infectivity and an approximately constant incubation period, we should expect to be able to distinguish different generations of the intra-household epidemic. There might be a binomial distribution of secondary cases resulting from contact with the primary case, to be followed later by another binomial distribution of tertiary cases amongst the susceptibles who had previously escaped, and so on. In his classic paper on this subject Greenwood (1931) introduced such a chainbinomial model for the investigation of measles epidemics, for which the assumptions made are thought to be approximately true. When the period of infectivity is more extended, as with scarlet fever or whooping cough for example, the stochastic model recently discussed by Bailey (1953) may be more appropriate. Greenwood examined data on the 1926 measles epidemic in St Pancras, and showed that, so far as the total number of cases in a household was concerned, the hypothesis of a simple binomial distribution was quite inadequate, while the distribution expected from the chain-binomial model gave a satisfactory fit to the observations. It is important to notice, however, that this material provided information only on the total size of the epidemic in a given household; no analysis of the individual links of the chain was possible. Now Wilson, Bennett, Allen & Worcester (1939), in their investigation of cases of measles occurring in Providence, Rhode Island, during 1929-34, were able to go a stage further and break down the data into its constituent parts. Of course it must be recognized that departures from a constant incubation period, or the possibility of multiple primary cases, give rise to certain difficulties in the so-called chaining of this kind of material. However, it seemed likely that in the present case this source of confusion would not be serious. Wilson et al. were able to show that in none of the groups of available data did the Greenwood model give an adequate description of the numbers of cases occurring in the separate generations through which the epidemic passed, though it did sometimes give a satisfactory fit to the distribution of the total number of cases in a household. In a later paper, Wilson (1947) considered a somewhat different approach. According

THE appearance of yet another paper on the sampling problem* directs attention to the success which has attended of recent years the efforts of workers in this field. The general problem considered by one group of workers is the following. Let there be given a population, supposed infinite in extent, but subject to this having any law of distribution with finite moments. It may be a population of one or manv variables. The population may be regarded as completely specified by a knowledge of all its characteristic parameters, which may be moment coefficients or semi-invariants, or expressible in terms of these. For a sample of size n drawn at randomi fiom this population we may calculate in sonme manner certain ftunctions which are to be regarded as estimates of the population moment coefficients, or semii-invariants. The simiultaneous distribuition in repeated samples of the various estimates will depend upon that of the parent population, and the problem I wish to take up deals with the determinationi of the moment coefficients, or sei-ni-invariants, of this simultaneous distribution. Prior to 1928 certain individual results oinly had been worked out; in that year two independent papers of great importance appeared. R. A. Fishert showed that if we define as estimates of the population semi-invariants (ic,.) certain functionis (kr) of the sample observations by means of the simple propeity that the mean value of kr for an infinite number of samples is to be K4++, then the semi-invariants of the simultaneous distribution of the k's are peculiarly simnple in form, compared to analogouis expressions derived in other ways. These semi-invariants can be derived algebraically, or more simply by following out certain straightforward combinatorial rules. Fisher's paper marked a great advance in showing the possibility of beginning a systematic tabulation of the required formtulae, a thing that had not before been possible. In his paper all formulae up to the tenth degree were given, together with a numniber of special interest of the twelfth degree. In addition the paper showed how the methods could readily be applied to multivariate populations, and a number of the more general fornmulae were given.

IN a first biometric study of Vespa vulgarTis made two years ago* its authors compared the size, variability and correlation of wing parts of queens, drones and workers from a single nest, which contained 129 queens, 150 drones and several hundred workers. In the present study the same measurements have been repeated on 188 queens, presumably belonging to many nests and collected in a different district. The nest of the first paper wvas taken in the autumn of 1905 in the neighbourhood of Godalming. The queens for this second investigation were obtained in the spring, of 1908 in the neighbourhood of Gerard's Cross. As soon as the first queen wasp appeared, a reward of ld. was offered for each queen wasp brought, and conisiderably over 200 were thus rapidly collected. Of this number certain individuals had to be rejected because their wings were damaged; a few othiers were rejected because their colour raised suspicion that they had been thrown in from an old nest. The specimens came in small numbers, sometimes one at a time, often alive, and there is no reason to doubt that thev represent a genuine sample of the queen wasps of the autunin of 1907, which survived the winter and were starting in the spring of 1908 to establish their nests.

1. THE relation between the two characters in a sample, mean Z and standard deviation a(or variance ar2), is a very interesting bne as several methods of judging whether a certain sample has been taken from a given population, or if two different samples have been taken from the same population, are based on the comparison of certain functions of x with certain functions of a. For the case where the original frequency distribution is normal, the question has been largely discussed, in particular in the paper of "Student"* in which the

One of the central problems in the theory of queues has been the study of the singleserver queue with various assumptions about the probabilistic nature of the input and service processes. In particular, the queue GI1//1 (in the notation of Kendall (1953)) has been studied by Lindley (1952), and a complete formal solution, under mild analytic conditions, for the equilibrium waiting-time distribution, has been obtained by Smith (1953). This solution is not, however, easy to evaluate except for very special distributions of interarrival and service times. The present paper is concerned with the mean and variance of the waiting time, which are quantities of some practical importance. Inequalities are derived for these moments, both in equilibrium and non-equilibrium conditions. Thus we consider a queueing system in which customers C0, C1, ... arrive and are attended to by a single server in the order of their arrival. The customer Cn waits in the queue for a time Wn, and is then served in a further time 8nWe denote by tn the time elapsing between the arrivals of successive customers Cn, Cn+?, and write n-i UnSntn Un E Ur. (1) r=O

In this paper we provide numerical solutions to the following decision problem. There are two terminal decisions and the states of the world are indexed by a parameter p, 0 < p < 1. The losses (or utilities) are linear in p. It is possible to observe independent and identically distributed random variables x1, x2, .... The probability that any xi = 1 is p, if not xi = 0. The cost of observing any xi is a constant. There exists a prior distribution of p which belongs to the beta family. A decision may be made after any number (including zero) of the random variables have been observed either to make one of the two terminal decisions or to observe a further xi. What is the optimum decision function ? The reason for considering this problem is that there are many situations in science and technology where it is possible to observe whether (xi = 1) or not (xi = 0) items possess a certain attribute and where it is necessary to make one of two decisions whose outcomes depend on the proportion p of items possessing the attribute. Under certain conditions the best way to make the decision is to proceed sequentially, that is by looking at the items one at a time, and after each inspection deciding whether to examine further items or to make a terminal decision. After xl, ..., Xn have been observed, and writing r = Exi, the likelihood is proportional to pr(l p)n-r. Consequently, it is simplest to use the beta prior distribution (see Raiffa & Schlaifer, 1961). The situation where the utilities are linear also recommends itself for first study. Possible applications of our results are to sampling inspection problems in industry, medical trials, sociological studies and other fields.

SUMMARY The general expressions obtained by Pillai (1965) for approximating at the upper end to the C.D.F. of the largest of s characteristic roots of a matrix jointly distributed according to the Fisher-Girshick-Hsu-Roy distribution, have been used to compute upper 5 and 1 % points of the largest root for s up to 20, of which those for s = 14, 16, 18 and 20 are presented in this paper. Methods of interpolation for obtaining such percentage points for intermediate values of s have been suggested and errors of interpolation and approximation discussed.

The cumulative distribution function (C.D.F.) of the largest characteristic root of a matrix in multivariate analysis has been studied by Pillai (1954, 1956a, 1957, 1960 and 1964) with a view to obtaining an approximation to this C.D.F. useful for computing the upper percentage points. The approach, so far, has been to approximate the C.D.F. of the largest root for each value of the number, s, of non-null roots from two to seven. In this paper, general expressions are given for the first time, approximating at the upper end the C.D.F. of the largest of s non-null characteristic roots. Further, these expressions are used to compute the upper 5 and 1 % points of the largest root for s = 8, 9 and 10. It may be pointed out that Roy (1945, 1953, 1957) has shown that tests of certain hypotheses in multivariate analysis and associated confidence interval estimation can be based on the extreme characteristic roots. The monotonic character of the power functions of two multivariate tests using the largest characteristic root has been demonstrated by several authors (Roy & Mikhail, 1961; Das Gupta, Anderson & Mudholkar, 1964; Anderson & Das Gupta, 1964).

IN a recent paper* Professor Pearson shows that where one character is in multiple graded grouping and the other in alternative categories, greater or less than a given magnitude, the correlation coefficient admits of simple expression; the assumptions being that the unmeasured character, B, has a normnal distribution and that the measured character, A, has linear regression upon B. Under these conditions the data required are the numerical ratio of the alternative groups, the standard deviation of the measured character and the deviation from the general mean of this character of the mean of one of the groups.

To calculate maximum-likelihood estimates of the mean and standard deviation of a normally distributed population from doubly truncated or from doubly censored samples, it is necessary to solve simultaneously a pair of rather complex non-linear estimating equations. Although solutions can be approximated by straightforward iterative procedures, the calculations often become tedious and time-consuming. This paper is concerned with reducing computational labour required in obtaining these solutions. For use in the doubly truncated case, a chart which permits direct reading of the standardized terminals with an accuracy of from three to five units in the second decimal is included. For use in both the doubly truncated and doubly censored cases, an iterative procedure is devised, which, for a specified degree of accuracy, appears to require less computational effort than similar procedures previously proposed. For use in singly censored cases, where only one estimating equation is involved, a chart which permits direct reading of estimates of the standardized terminal with a degree of accuracy comparable to that possible in the doubly truncated case is presented. Since tables recently published by Cohen & Woodward (1953) reduce the calculation of estimates from singly truncated norinal samples to the simple task of interpolating between table entries, that case is considered here only by reference.

The problem of two samples has been considered by several authors from various points of view. Most of this work is devoted to testing hypotheses concerning differences in location of two populations. A large number of non-parametric tests have been suggested. Particular mention may be made of the Wilcoxon test which has been considered by several authors especially Mann & Whitney (1947), Lehmann (1951), Sundrum (1953) and Mood (1954). Hodges & Lehmann (1956) have shown that the Wilcoxon test is a good competitor to the Student's t-test. However, very little has been done in the direction of testing differences in the dispersion of two populations. Recently, attempts have been made by several authors, especialy Mood (1954), Lehmann (1951), Sukhatme (1957, 1958a), Rosenbaum (1953) and Kamat (1956) to suggest suitable non-parametric tests for testing differences in dispersion. The tests proposed by the first three authors are based on what are known as generalized U statistics and the present author has recently examined the relative efficiency of these tests. However, nothing is known concerning the power of the tests suggested by the last two authors. This paper is a contribution in that direction and discusses the power of these two tests. 2. WILKS-ROSENBAUM TEST The test is based on the number of observations of one sample, say the X's, which lie outside the extreme values of the other sample. We reject the hypothesis if this number is

SUMMARY Although the conditions for application of the technique of empirical Bayes estimation of the parameter of a distribution may exist, the form of the prior distribution is generally unknown. This often creates major difficulties in the determination of empirical Bayes estimators. The simple device of approximating the prior distribution by a step function is used to overcome this problem and to obtain smooth empirical Bayes estimators. This paper supplements earlier work by considering continuous distributions and multiple past and current observations. The effectiveness of the proposed smooth empirical Bayes estimators is examined, and the results indicate that they can be substantially better than optimum non-Bayes estimators, even when the amount of past data is small.

1. Summary. Procedures for estimating the relative efficiency of certain designs have been considered by Yates (1935), Cochran & Cox (1957), and Kempthorne (1952, 1955). In this paper the efficiency of blocking in general is considered. A general formula for any incomplete block design of fairly general form is obtained. 2. Introduction. In many experiments one has the option of choosing from a wide class of designs ranging from designs involving no blocking to designs involving two-way classification of experimental units. A measure of the efficieiAcy of one design relative to another is the inverse ratio of the variances with which treatment differences are estimated in the two designs. Given that design A has been used, estimation of its efficiency relative to some other design, B say, requires that we estimate the error variance of design B from the data obtained from design A. A common elementary situation is that a randomized block design has been used and it is desired to estimate the error variance had a completely randomized design been used. The formula for this purpose is well known, but we wish to summarize a few of the salient points of the randomization theory upon which it is based. Throughout we shall concern ourselves only with variability over which we randomize. Consider the conceptual population of yields that would be obtained if every treatment could be applied to every plot. Denote the yield with treatment k ( = 1, 2, ..., t) on plot

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).

is being increasingly used in statistical work in many branches of biology. Various numerical methods of fitting this curve have been suggested in recent years for cases in which the y's are independent and have equal variances and the x's take the n values 0, 1, 2, ..., n-1. Stevens (1951) described a fully efficient least squares method for estimating a, ,8 and p, and the errors of these estimates. The method is one of successive approximation and requires a reasonably accurate initial estimate of p. It works well on an electronic computer. He also provided tables for n = 5, 6 and 7, since extended by S. Lipton, which are very useful when the computations are carried out on a desk machine. Pimentel Gomes (1953) provided further tables for n = 4 and n = 5 which permit the least squares estimates of p to be obtained more rapidly than by the iterative procedure. In a previous paper I pointed out that the least squares estimates of p, r' say, are of the form n-1 EWX (A)YX r