This paper contains details of the results announced in the abstract by the authors (1962). Techniques are developed for proving local minimax and "type $D$" properties and asymptotic (that is, far in distance from the null hypothesis) minimax properties in complex testing problems where exact minimax results seem difficult to obtain. The techniques are illustrated in the settings where Hotelling's $T^2$ test and the test based on the squared sample multiple correlation coefficient $R^2$ are customarily employed.

Let $R$ be the set of real numbers, $\mathscr{B}_1$ the set of Borel sets of $R$, and $\mu$ a $\sigma$-finite nonnegative measure on $\mathscr{B}_1$. Let $\Omega$ be an open real number interval (which may be infinite). Throughout we consider a Koopman-Darmois family \begin{equation*}\tag{1}\{h(\theta) \exp (\theta x), \theta \varepsilon \Omega\}\end{equation*} of generalized probability density functions on the measure space $(R, \mathscr{B}_1, \mu)$. We consider one sided tests $T$ of the hypothesis $\theta < 0$ against the alternative $\theta > 0$. In general, in this paper, $T$ will be a sequential procedure. Associated with $T$ is a stopping variable $N$ (mention of the dependence of $N$ on $T$ is usually omitted). $N \geqq 0. N = n$ means that sampling stopped after $n$ observations and a decision was made. In this context we consider $\infty$ to be an integer, and $N = \infty$ means that sampling does not stop. In the discussion of Section 1 we will assume that if $\theta \varepsilon \Omega$ and $\theta \neq 0$ then $P_\theta(N < \infty) = 1$, that is, sampling stops with probability one. The reason for the exclusion of $\theta = 0$ will become apparent in Section 1. We will be concerned with two events, decide $\theta < 0$, and, decide $\theta > 0$. The main result of this paper may be stated as follows. Theorem 1. Suppose $(R, \mathscr{B}_1, \mu), \Omega$, and $\{h(\theta) \exp (\theta x), \theta \varepsilon \Omega\}$ are as described above. Define \begin{align*}\mu_\theta = \int^\infty_{-\infty} h(\theta)x \exp(\theta x)\mu(dx), \\ \tag{2} \\ \sigma^2 = \int^\infty_{-\infty} h(0)x^2\mu(dx),\end{align*} and assume $\mu_0 = 0$. Suppose $0 < \alpha < 1$ and $0 < \beta < 1$ and \begin{equation*}\tag{3}\sup_{\theta > 0} P_\theta (\text{ decide } \theta < 0) \leqq \beta;\quad \sup_{\theta < 0} P_\theta (\text{ decide } \theta > 0) \leqq \alpha.\end{equation*} Then \begin{align*}\lim \sup_{\theta \rightarrow 0+} \mu^2_\theta|\log|\log\mid\mu_\theta|\|^{-1}E_\theta N \geqq 2\sigma^2 P_0(N &= \infty); \\ \tag{4} \lim \sup_{\theta \rightarrow 0-} \mu^2_\theta|\log|\log\mid\mu_\theta|\|^{-1}E_\theta N \geqq 2\sigma^2 P_0(N &= \infty).\end{align*} If $\alpha + \beta < 1$ there is a generalized sequential probability ratio test $T$ with stopping variable $N$ such that for the test $T$, \begin{equation*}\tag{5}P_0(N = \infty) = 1 - (\alpha + \beta);\quad (3) \text{ holds };\end{equation*} for the test $T$, \begin{equation*}\tag{6}\lim_{\theta \rightarrow 0} \mu^2_\theta|\log|\log\mid\mu_\theta|\|^{-1}E_\theta N = 2\sigma^2 P_0(N = \infty).\end{equation*} For all tests $T$, if $P_0(N = \infty) > 0$ then $\lim_{\theta \rightarrow 0} \theta^2E_\theta N = \infty$. In Section 1, (7) and (8), it is shown that $P_0(N = \infty) \geqq 1 - \alpha - \beta$. Consequently the relations (4) and (5) of Theorem 1 are not vacuous. We were led to formulate Theorem 1 by a problem of constructing bounded length confidence intervals. The relationship is explained in Section 2. The proof of Theorem 1 is given in Section 3.

This paper contains a new approach toward a theory of robust estimation; it treats in detail the asymptotic theory of estimating a location parameter for contaminated normal distributions, and exhibits estimators--intermediaries between sample mean and sample median--that are asymptotically most robust (in a sense to be specified) among all translation invariant estimators. For the general background, see Tukey (1960) (p. 448 ff.) Let $x_1, \cdots, x_n$ be independent random variables with common distribution function $F(t - \xi)$. The problem is to estimate the location parameter $\xi$, but with the complication that the prototype distribution $F(t)$ is only approximately known. I shall primarily be concerned with the model of indeterminacy $F = (1 - \epsilon)\Phi + \epsilon H$, where $0 \leqq \epsilon < 1$ is a known number, $\Phi(t) = (2\pi)^{-\frac{1}{2}} \int^t_{-\infty} \exp(-\frac{1}{2}s^2) ds$ is the standard normal cumulative and $H$ is an unknown contaminating distribution. This model arises for instance if the observations are assumed to be normal with variance 1, but a fraction $\epsilon$ of them is affected by gross errors. Later on, I shall also consider other models of indeterminacy, e.g., $\sup_t |F(t) - \Phi(t)| \leqq \epsilon$. Some inconvenience is caused by the fact that location and scale parameters are not uniquely determined: in general, for fixed $\epsilon$, there will be several values of $\xi$ and $\sigma$ such that $\sup_t|F(t) - \Phi((t - \xi)/\sigma)| \leqq \epsilon$, and similarly for the contaminated case. Although this inherent and unavoidable indeterminacy is small if $\epsilon$ is small and is rather irrelevant for practical purposes, it poses awkward problems for the theory, especially for optimality questions. To remove this difficulty, one may either (i) restrict attention to symmetric distributions, and estimate the location of the center of symmetry (this works for $\xi$ but not for $\sigma$); or (ii) one may define the parameter to be estimated in terms of the estimator itself, namely by its asymptotic value for sample size $n \rightarrow \infty$; or (iii) one may define the parameters by arbitrarily chosen functionals of the distribution (e.g., by the expectation, or the median of $F$). All three possibilities have unsatisfactory aspects, and I shall usually choose the variant which is mathematically most convenient. It is interesting to look back to the very origin of the theory of estimation, namely to Gauss and his theory of least squares. Gauss was fully aware that his main reason for assuming an underlying normal distribution and a quadratic loss function was mathematical, i.e., computational, convenience. In later times, this was often forgotten, partly because of the central limit theorem. However, if one wants to be honest, the central limit theorem can at most explain why many distributions occurring in practice are approximately normal. The stress is on the word "approximately." This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): What happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance: seemingly quite mild deviations may already explode its variance. Tukey and others proposed several more robust substitutes--trimmed means, Winsorized means, etc.--and explored their performance for a few typical violations of normality. A general theory of robust estimation is still lacking; it is hoped that the present paper will furnish the first few steps toward such a theory. At the core of the method of least squares lies the idea to minimize the sum of the squared "errors," that is, to adjust the unknown parameters such that the sum of the squares of the differences between observed and computed values is minimized. In the simplest case, with which we are concerned here, namely the estimation of a location parameter, one has to minimize the expression $\sum_i (x_i - T)^2$; this is of course achieved by the sample mean $T = \sum_i x_i/n$. I should like to emphasize that no loss function is involved here; I am only describing how the least squares estimator is defined, and neither the underlying family of distributions nor the true value of the parameter to be estimated enters so far. It is quite natural to ask whether one can obtain more robustness by minimizing another function of the errors than the sum of their squares. We shall therefore concentrate our attention to estimators that can be defined by a minimum principle of the form (for a location parameter): $T = T_n(x_1, \cdots, x_n) minimizes \sum_i \rho(x_i - T),$ \begin{equation*} \tag{M} where \rho is a non-constant function. \end{equation*} Of course, this definition generalizes at once to more general least squares type problems, where several parameters have to be determined. This class of estimators contains in particular (i) the sample mean $(\rho(t) = t^2)$, (ii) the sample median $(\rho(t) = |t|)$, and more generally, (iii) all maximum likelihood estimators $(\rho(t) = -\log f(t)$, where $f$ is the assumed density of the untranslated distribution). These ($M$)-estimators, as I shall call them for short, have rather pleasant asymptotic properties; sufficient conditions for asymptotic normality and an explicit expression for their asymptotic variance will be given. How should one judge the robustness of an estimator $T_n(x) = T_n(x_1, \cdots, x_n)$? Since ill effects from contamination are mainly felt for large sample sizes, it seems that one should primarily optimize large sample robustness properties. Therefore, a convenient measure of robustness for asymptotically normal estimators seems to be the supremum of the asymptotic variance $(n \rightarrow \infty)$ when $F$ ranges over some suitable set of underlying distributions, in particular over the set of all $F = (1 - \epsilon)\Phi + \epsilon H$ for fixed $\epsilon$ and symmetric $H$. On second thought, it turns out that the asymptotic variance is not only easier to handle, but that even for moderate values of $n$ it is a better measure of performance than the actual variance, because (i) the actual variance of an estimator depends very much on the behavior of the tails of $H$, and the supremum of the actual variance is infinite for any estimator whose value is always contained in the convex hull of the observations. (ii) If an estimator is asymptotically normal, then the important central part of its distribution and confidence intervals for moderate confidence levels can better be approximated in terms of the asymptotic variance than in terms of the actual variance. If we adopt this measure of robustness, and if we restrict attention to ($M$)-estimators, then it will be shown that the most robust estimator is uniquely determined and corresponds to the following $\rho:\rho(t) = \frac{1}{2}t^2$ for $|t| < k, \rho(t) = k|t| - \frac{1}{2}k^2$ for $|t| \geqq k$, with $k$ depending on $\epsilon$. This estimator is most robust even among all translation invariant estimators. Sample mean $(k = \infty)$ and sample median $(k = 0)$ are limiting cases corresponding to $\epsilon = 0$ and $\epsilon = 1$, respectively, and the estimator is closely related and asymptotically equivalent to Winsorizing. I recall the definition of Winsorizing: assume that the observations have been ordered, $x_1 \leqq x_2 \leqq \cdots \leqq x_n$, then the statistic $T = n^{-1}(gx_{g + 1} + x_{g + 1} + x_{g + 2} + \cdots + x_{n - h} + hx_{n - h})$ is called the Winsorized mean, obtained by Winsorizing the $g$ leftmost and the $h$ rightmost observations. The above most robust ($M$)-estimators can be described by the same formula, except that in the first and in the last summand, the factors $x_{g + 1}$ and $x_{n - h}$ have to be replaced by some numbers $u, v$ satisfying $x_g \leqq u \leqq x_{g + 1}$ and $x_{n - h} \leqq v \leqq x_{n - h + 1}$, respectively; $g, h, u$ and $v$ depend on the sample. In fact, this ($M$)-estimator is the maximum likelihood estimator corresponding to a unique least favorable distribution $F_0$ with density $f_0(t) = (1 - \epsilon)(2\pi)^{-\frac{1}{2}}e^{-\rho(t)}$. This $f_0$ behaves like a normal density for small $t$, like an exponential density for large $t$. At least for me, this was rather surprising--I would have expected an $f_0$ with much heavier tails. This result is a particular case of a more general one that can be stated roughly as follows: Assume that $F$ belongs to some convex set $C$ of distribution functions. Then the most robust ($M$)-estimator for the set $C$ coincides with the maximum likelihood estimator for the unique $F_0 \varepsilon C$ which has the smallest Fisher information number $I(F) = \int (f'/f)^2f dt$ among all $F \varepsilon C$. Miscellaneous related problems will also be treated: the case of non-symmetric contaminating distributions; the most robust estimator for the model of indeterminacy $\sup_t|F(t) - \Phi(t)| \leqq \epsilon$; robust estimation of a scale parameter; how to estimate location, if scale and $\epsilon$ are unknown; numerical computation of the estimators; more general estimators, e.g., minimizing $\sum_{i < j} \rho(x_i - T, x_j - T)$, where $\rho$ is a function of two arguments. Questions of small sample size theory will not be touched in this paper.

For testing the equality of $c$ continuous probability distributions on the basis of $c$ independent random samples, the test statistics of the form $\mathscr{L} = \sum^c_{j = 1} m_j\lbrack(T_{N,j} - \mu_{N,j})/A_N\rbrack^2$ are considered. Here $m_j$ is the size of the $j$th sample, $\mu_{N,j}$ and $A_N$ are normalizing constants, and $T_{N,j} = (1/m_j) \sum^N_{i = 1} E_{N,i}Z^{(j)}_{N,i}$ where $Z^{(j)}_{N,i} = 1$, if the $i$th smallest of $N = \sum^N_{j = 1} m_j$ observations is from the $j$th sample and $Z^{(j)}_{N,i} = 0$ otherwise. Sufficient conditions are given for the joint asymptotic normality of $T_{N,j}; j = 1, \cdots, c$. Under suitable regularity conditions and the assumption that the $i$th distribution function is $F(x + \theta_i/N^{\frac{1}{2}})$, the limiting distribution of $\mathscr{L}$ is derived. Finally, the asymptotic relative efficiencies in Pitman's sense of the $\mathscr{L}$ test relative to some of its competitors viz. the Kruskal-Wallis $H$ test (which is a particular case of the $\mathscr{L}$ test) and the classical $F$ test are obtained and shown to be independent of the number $c$ of samples.

Two nonparametric tests presently available for testing the equality of several treatments (varieties, objects etc.) on the basis of paired-comparisons are: (a) the Bradley-Terry [4] likelihood ratio test; (b) Durbin's [7] test (which, in fact, covers the general balanced incomplete block design). These tests, however, were proposed for the case when no meaningful measurements on the quality of treatments are possible. Instead, one can merely decide for each individual comparison which treatment to prefer. Both these tests are, thus, instances where only signs of the comparison differences are involved, and as such can be regarded as generalizations of the Sign-test. The large sample properties of these tests have been discussed by Bradley [3] and by Van Elteren and Noether [17] respectively. As shown in the latter paper, both these tests have asymptotic (Pitman) efficiency equal to $2/\pi$ relative to the $F$-test (under normality). One can reasonably hope to improve this efficiency by taking into consideration the magnitudes of the observed comparison differences, when they are available. A test of this nature, based on a generalization of the Wilcoxon-one-sample ranking procedure, is proposed and investigated below for the case when all comparisons are performed under the same experimental conditions (Section 2). The asymptotic distribution of the test statistic proposed is obtained using the results of Godwin and Zaremba [8] and Konijn [12] (Section 3). It is shown that the asymptotic efficiencies of this test relative to the Durbin and the Bradley-Terry tests and the corresponding $F$-test are independent of the number of treatments involved. These results are also extended to the case of non-uniformity of experimental conditions (Section 4). (The attention of the reader is also drawn to a rank procedure suggested by Hodges and Lehmann [11] for the general incomplete block design which, in particular, is also applicable to the present problem. However, the efficiency of this procedure has so far not been fully investigated.)

This paper is concerned with the problem of testing the independence of two random variables $X, Y$ on the basis of a random sample, $(X_1, Y_1), (X_2, Y_2), \cdots, (X_N, Y_N)$. The joint distribution function $H$ and, consequently, the marginal distribution functions $F$ and $G$ are assumed to be absolutely continuous. The hypothesis to be tested may be stated as $H(x, y) = F(x)G(y)$. The usual parametric test of this hypothesis is based on the sample correlation coefficient. Several nonparametric tests have been proposed and studied by Kendall [11], Hoeffding [8], [9], Blomqvist [1], and others. Konijn [13] has investigated the asymptotic power properties of some of these tests. The class of rank tests for independence to be considered in this paper is based on the test statistics of the form \begin{equation*}\tag{1.1}T_N = N^{-1}\sum^N_{i = 1} E_{N,r_i}E'_{N,s_i}Z'_{N,r_i}Z'_{N,s_i}\end{equation*} where $\{E_{N,i}\}, \{E'_{N,i}\}, i = 1, 2, \cdots, N$, are two sets of constants satisfying certain restrictions to be stated below, and $Z_{N,r_i} = 1 (Z'_{N,s_i} = 1)$ when $X_i(Y_i)$ is the $r_i$th ($s_i$th) smallest of the $X$'s ($Y$'s) and $Z_{N,r_i} = 0 (Z'_{N,s_i} = 0)$ otherwise. By taking $E_{N,r_i}(E'_{N,s_i})$ to be the expected value of the $r_i$th ($s_i$th) standard normal order statistic from a sample of size $N$, we get the normal scores test statistic which belongs to the class of $c_1$-statistics considered by Fisher and Yates [4], Hoeffding [10] and Terry [17]. If we put $E_{N,r_i} = r_i$ and $E_{N,s_i} = s_i$, the resulting test statistic is equivalent to the Spearman rank correlation statistic. The normal scores test is shown to be (a) the locally most powerful rank test and (b) asymptotically as efficient as the parametric correlation coefficient $\mathscr{R}$-test for the alternatives (4.1) and (4.2) when the underlying distributions are normal. It is at least as efficient as the $\mathscr{R}$-test when the alternative belongs to the class (4.1) and when $H$ satisfies the restrictions stated in Theorem 3. Tables 1 and 2 give the exact null distribution and some critical values of the normal scores statistic for $N \leqq 6$. Table 2 provides a comparison of the $t$-approximation with the exact distribution.

The object of this article is the analysis of the concept of distribution-freeness. In Section 2 the common types of distribution-free (DF) statistics are defined and related. In Section 3 a generalization of the invariance principle is employed to construct DF statistics for classes of cpfs generated by groups of transformations of the sample space. It is shown in Section 4 that a function of several DF statistics is not necessarily DF, but that independence guarantees the desired result. The existence of a DF statistic is shown, in Section 5, to be equivalent to the existence of a suitable partition of the sample space. Several open problems are mentioned in Section 6, the final section.

Asymptotic expansions are derived for the behavior of the optimal sequential test of whether the unknown drift $\mu$ of a Wiener-Levy process is positive or negative for the case where the process has been observed for a long time. The test is optimal in the sense that it is the Bayes test for the problem where we have an a priori normal distribution of $\mu$, the regret for coming to the wrong conclusion is proportional to $|\mu|$, and the cost of observation is constant per unit time. The Bayes procedure is then compared with the best sequential likelihood ratio test.

In this paper sequential procedures are given for selecting the normal population with the greatest mean when (a) the $k$ populations have a common known variance or (b) the $k$ populations have a common but unknown variance, so that in each case the probability of making the correct selection exceeds a specified value when the greatest mean exceeds all other means by at least a specified amount. The procedures in the present paper all have the property that inferior populations can be eliminated from further consideration as the experiment proceeds.

Let the random vector $X = (X_1 \cdots X_p)'$ have a multivariate normal distribution with unknown mean $\xi = (\xi_1 \cdots \xi_p)'$ and unknown nonsingular covariance matrix $\Sigma$. Write $\Sigma^{-1}\xi = \Gamma = (\Gamma_1 \cdots \Gamma_p)'$. The problem considered here is that of testing the hypothesis $H_0 : \Gamma_{q + 1} = \cdots = \Gamma_p = 0$ against the alternative $H_1 : \Gamma_{p' + 1} = \cdots = \Gamma_p = 0$ when $p \geqq p' > q$ and $\xi, \Sigma$ are both unknown. This problem arises in discriminating between two multivariate normal populations with the same unknown covariance matrix when one is interested to test whether the variables $X_{q + 1} \cdots X_{p'}$ contribute significantly to the discrimination. For a comprehensive treatment of this subject, the reader is referred to Rao (1952), Chapter 7. In this paper we will find the likelihood ratio test of $H_0$ against $H_1$ and show that this test is uniformly most powerful similar invariant. The problem of testing $H_0$ against $H_1$ remains invariant under the groups $G_1$ and $G_2$ where $G_1$ is the group of $p' \times p'$ non-singular matrices $g = \begin{pmatrix}g_{11} & 0\\g_{22} & g_{22}\end{pmatrix}$ (with $g_{11}$ a $q \times q$ matrix) which transform the coordinates $X_1 \cdots X_{p'}$ of $X$ and $G_2$ is the group of translations of the coordinates $X_{p' + 1} \cdots X_p$ of $X$. We may restrict our attention to the space of the sufficient statistic $(\bar X, S)$ of $(\xi, \Sigma)$. A maximal invariant under $G_1$ and $G_2$ in the space of $(\bar X, S)$ is $R = (R_1, R_2)'$, and a corresponding maximal invariant in the parametric space of $(\xi, \Sigma)$ is $\delta = (\delta_1, \delta_2)'$, where $R_i \geqq 0, \delta_i \geqq 0$ are defined in Section 2. In Section 1, we will find the likelihood ratio test of $H_0$ against $H_1$ in the usual way. The likelihood ratio test is invariant under all transformations which keep the problem invariant, and hence is a function only of $R$. In Section 2, we will find the joint density of $R_1$ and $R_2$ under the hypothesis and under the alternatives and then follow Neyman's approach of invariant similar regions to show that the likelihood ratio test in this case is uniformly most powerful similar invariant. In terms of maximal invariants, the above problem reduces to that of testing $H_0 : \delta_2 = 0, \delta_1 > 0$ against the alternative $H_1 : \delta_2 > 0, \delta_1 > 0$. According to a Fisherian philosophy of statistical inference applied to invariant procedures, it is reasonable to think of $R_1$ as giving information about the discriminating ability of the set of variables $(X_1 \cdots X_q)$, but no information about parameters governing additional discriminating ability from variables $X_{q + 1} \cdots X_{p'}$. Thus Fisher might call $R_1$ ancillary for the problem at hand and condition on it. We are not concerned here with the philosophical issues of statistical inference; instead, we will find (in Section 3) the distribution of the likelihood ratio conditional on $R_1$ which forms the basis of inference in a Fisherian approach. It will be shown that in this conditional situation, the likelihood ratio test is uniformly most powerful invariant. A more general statement of this same problem is to find the likelihood ratio test of the hypothesis $H'_0 : \Gamma \varepsilon \mathscr{Z}'$ that $\Gamma$ belongs to $\mathscr{Z}'$ against the alternative $H'_1 : \Gamma \varepsilon \mathscr{Y}'$ that $\Gamma$ belongs to $\mathscr{Y}'$, when $\xi, \Sigma$ are both unknown and $\mathscr{Z}' \subset \mathscr{Y}'$ are linear sub-spaces of the adjoint space $\mathscr{X}'$ of the space of $X$'s, and are of dimensions $q$ and $p'$ respectively. This problem can be easily reduced to that above by a proper choice of coordinate system, depending on the particular forms of $\mathscr{Z}'$ and $\mathscr{Y}'$. One could have worked with this general formulation instead of that above but the author did not find it convenient for computational purposes. As a corollary, if $q = 0$ then $H_0$ falls back to the usual null hypothesis of multivariate analysis of variance. It is easy to see that the likelihood ratio test for $q = 0$ reduces to the usual Hotelling's $T^2$ test which is uniformly most powerful invariant (Lehmann (1959)). Fisher (1938) has dealt with a particular case of the general formulation where $\mathscr{Z}'$ is the one-dimensional linear sub-space of $\mathscr{X}'$, and a test based on a discriminant function was suggested by him. The problem of testing $H_0$ against $H_1$ has been dealt with by Rao (1949) and a test depending on the ratio of Mahalanobis' $D^2$ statistics based on the first $q$ and $p'$ components of $X$ (which is related to Fisher's discriminant function in a simple manner) was suggested by him. It will be seen that both the tests are the likelihood ratio test.

A pair of test statistics is proposed for the null hypothesis $\mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$ when the data consists of a sample from each of the $p$-variate normal distributions $N(\mathbf{u}_1, \mathbf{\Sigma}_1)$ and $N(\mathbf{u}_2, \mathbf{\Sigma}_2)$. These tests are motivated in Section 1 and defined explicitly in Section 2. Section 3 proves a theorem which includes the null hypothesis distribution theory of the tests. Section 4 gives some details of the computation of the test statistics. An appendix describes the shadow property of concentration ellipsoids which facilitates the geometrical discussion earlier in the paper.

The test procedures, invariant under certain groups of transformations [4], for testing a set of multivariate linear hypotheses in the linear normal model depend on the characteristic roots of a random matrix. The power function of such a test depends on the characteristic roots of a corresponding population matrix as parameters; these roots may be regarded as measures of deviation from the hypothesis tested. In this paper sufficient conditions on the procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test [1], Lawley-Hotelling trace test [1], and Roy's maximum root test [6] satisfy these conditions. The monotonicity of the power function of Roy's test has been shown by Roy and Mikhail [5] using a geometrical method.

For testing independence between two sets of normally distributed variates we consider the class of test procedures which are invariant under certain groups of transformations and depend only on the sample canonical correlation coefficients [1]. The power function of such a test depends only on the population canonical correlation coefficients as parameters, which may be regarded as measures of deviation from the hypothesis. In this paper sufficient conditions on the invariant procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test [1] and Roy's maximum root test [5] satisfy these conditions. In [4] only the unbiasedness of the maximum root test was proved, although the authors claimed to prove the monotonicity property.

In this paper we have investigated the effects on the operating characteristics of a test for the scale parameter of the exponential distribution on the assumption that the sample is from a "complete" exponential population when in reality it is known to have come from a truncated exponential population. The results derived are valid for samples of any size.

A method is described for computing the asymptotic variance of the maximum likelihood correlation estimator $\tilde{\rho}$, which uses the number of observations in symmetrically placed corner regions, ignoring a middle section. It is shown that the optimum location of the regions varies with the true value of the correlation. Tables and figures are presented showing the optimal locations and the efficiency of the estimation procedure for various locations under two conditions: one, where cost depends on total sample size and two, where cost depends on the number of observations in the corner regions. In the second case, the method discussed is shown to be able to attain any given precision more cheaply than estimation by the product moment correlation coefficient.

The problem of finding unbiased ratio estimators of the population total of some character with the help of an auxiliary character has drawn much attention in recent years. Some references to this are given at the end. Under commonly adopted sampling schemes Hartley and Ross (1954), Goodman and Hartley (1958), Mickey (1959) and others derived certain unbiased ratio type estimators of the population total, while Lahiri (1951), Midzuno (1952), Des Raj (1954), Nanjamma, Murthy, and Sethi (1959) and others gave modifications of certain sampling schemes under which the ratio estimators of current type were unbiased. The first group of authors was primarily concerned with getting new types of unbiased ratio estimators under common sampling schemes. However, the second group was concerned with introducing little modifications in certain sampling schemes so that the usual ratio estimators of these sampling schemes become unbiased under the modified sampling schemes. Certain extensions in the latter direction are given in this paper. No comparison concerning the relative merits and demerits of the two methods of getting unbiased ratio type estimators is attempted here. Nanjamma, Murthy, and Sethi (1959) have given a general procedure of unbiased estimation of certain type of parameters such as population total and variance etc., and have shown how a given sampling scheme can be modified to make ratio estimators of such parameters unbiased. The modification can be applied to most of the sampling schemes commonly met in practice. In this paper it is shown that if in these modified sampling schemes a sufficient statistic is available [as is usually the case in with replacement sampling schemes, reference Basu (1958) and Pathak (1962(a), (b))] and if the ratio estimator does not depend on the sufficient statistic, it can be uniformly improved by Rao-Blackwell theorem. This result has been used to derive ratio estimators better than the ratio estimators given by Nanjamma, Murthy and Sethi. It is shown that in these modified sampling schemes the improved ratio estimator is the ratio of improved estimators of the numerator to the denominator derived under the original sampling scheme (without modification). Application of this result is given to some commonly used sampling schemes.

If $X$ and $Y$ are independent random variables having chi-square distributions with $n$ and $m$ degrees of freedom, respectively, then except for constants, $X/Y$ and $X/(X + Y)$ are distributed as $F$ and Beta variables. In the multivariate case, the Wishart distribution plays the role of the chi-square distribution. There is, however, no single natural generalization of a ratio in the multivariate case. In this paper several generalizations which lead to multivariate analogs of the Beta or $F$ distribution are given. Some of these distributions arise naturally from a consideration of the sufficient statistic or maximal invariant in various multivariate problems, e.g., (i) testing that $k$ normal populations are identical [1], p. 251, (ii) multivariate analysis of variance tests [9], (iii) multivariate slippage problems [4], p. 321. Although several of the results may be known as folklore, they have not been explicitly stated. Other of the distributions obtained are new. Intimately related to some of the distributional problems is the independence of certain statistics, and results in this direction are also given.

In testing hypotheses about an exponential distribution with probability density function $p(x; \theta, A) = (1/\theta)e^{-(x - A)/\theta},\quad \text{ for } x \geqq A,\\= 0,\quad \text{ for } x < A,$ where $\theta > 0$, the following questions arise: (1) Are certain linear forms in the order statistics of a random sample of size $n$ from this distribution distributed as chi-square random variables? (2) Are certain linear forms in the order statistics stochastically independent? The first two theorems in Section 2 answer these questions. As a consequence of these two theorems, several results follow which are similar to those pertaining to quadratic forms in normally distributed variables. The characterization theorem in Section 3 was suggested by a result for normally distributed variables. Lukacs [4] proved that if a random sample is taken from a continuous type distribution with finite variance, then the independence of the sample mean and the sample variance characterizes the normal distribution. That is, the independence of the estimates of the two parameters of the normal distribution characterizes that distribution. Now if $X_1 < X_2 < \cdots < X_n$ are the order statistics of a random sample from the exponential distribution $p(x; \theta, A)$, then $X_1$ and $(1/n) \sum^n_{i = 1} (X_i - X_1)$ are estimates of the parameters $A$ and $\theta$, respectively. In Section 3, we prove that the independence of these two statistics characterizes this exponential distribution.

A distribution of a sum of identically distributed Gamma-variables correlated according to an "exponential" autocorrelation law $\rho_{kj} = \rho^{|k - j|}(k, j = 1, \cdots n)$ where $\rho_{kj}$ is the correlation coefficient between the $k$th and $j$th random variables and $0 < \rho < 1$ is a given number is derived. An "approximate" distribution of the sum of these variables under the assumption that the sum itself is a Gamma-variable is given. A comparison between exact and approximate distributions for certain values of the correlation coefficient, the number of variables in the sum and the values of parameters of the initial distributions is presented.

The paper tabulates individual and cumulative terms of the logarithmic series probability distribution using the mean of the distribution as entry argument. The range of the table is $\mu = 1.1 (0.1) 2.0 (0.5) 5.0 (1.0) 10.0$. The tables are prefaced by introductory material, outlining the history and applications of the distribution, deducing generating functions and moments, and giving a method of obtaining the terms of logarithmic series distributions outside the range of those tabled.

A probability distribution function $F$ is said to be infinitely divisible if, for every integer $n$, there is a distribution function $F_n$ such that $F$ is the $n$-fold convolution of $F_n$. If $F$ is infinitely divisible its characteristic function can be written in the Khinchin-Levy canonical form: \begin{equation*}\tag{1}f(u) = \exp\big\{i\gamma u + \int^\infty_{-\infty}\big(e^{iux} - 1 - \frac{iux}{1 + x^2}\big)\frac{1 + x^2}{x^2} dG(x)\big\},\end{equation*} where $\gamma$ is a constant and $G$ is a bounded, non-decreasing function. Hartman and Wintner [2] proved that if $G$ is discrete, then the distribution function $F$ is pure, i.e., it is either absolutely continuous or discrete or continuous singular, and an example was given of each of these types of pure distributions which was determined by a discrete $G$. In the example of a discrete $G$ producing a continuous singular $F, G$ was given jumps of size $(1/N)^{2j}$ at $\pm 1/N^j, j = 1, 2, \cdots$, where $N$ is a positive integer $\geqq 2$. It was first proved that $F$ must be continuous; then it was proved that $f(u)$ does not converge to zero as $|u| \rightarrow \infty$, thus violating the conclusion of the Riemann-Lebesgue lemma. The main purpose of this paper is to give sufficient conditions that a continuous singular $F$ be obtained from a discrete $G$. These conditions are not too broad; for example, they do not include the example cited above. However, these conditions should be of considerable interest in that they are obtained by purely probabilistic methods, there being no use made of the Riemann-Lebesgue lemma, and thus supply more insight into the structure of continuous singular infinitely divisible distributions. In Section 2 a lemma is proved which might be of independent interest. This lemma is used to prove Theorem 1 in Section 3, which gives sufficient conditions that $F$ be continuous singular. In Section 4 a theorem is proved, giving sufficient conditions that every $m$-fold convolution of $F, F^{\ast m}$, be continuous singular.

A multivariate competition process (M.C.P.) is a stationary, continuous time Markov process whose state space is the lattice points of the positive orthant in $N$-dimensional space and whose transition probability matrix only allows jumps to certain nearest neighbors. As such it is the natural generalization of birth and death processes. In this paper we extend the results of Reuter [15] to obtain sufficient conditions for a M.C.P. to be regular, positive recurrent, absorbed with certainty, and to have finite mean absorption time. Some explicit examples are given and references to various applications indicated.

In an election return with two candidates $A$ and $B$, if $\alpha_r$ is the number of votes for $A$ in the first $r$ counted, $\beta_r$ the similar number for $B$, then $r$ is a $c$-lead position for $A$ if $\alpha_r > \beta_r + c - 1$. With final vote $(n, m) (n$ for $A, m$ for $B$), what is the number $l_j(n, m; c)$ of returns with $j c$-lead positions? Or, what is the enumerator $l_{n m}(x; c) = \sum l_j(n, m; c)x^j$ of election returns by number of lead positions? For $c = 0, \pm 1, \pm 2, \cdots$ it is shown that all enumerators may be expressed in terms of $l_{n m}(x; 0)$ and $l_{n m}(x; 1)$, which are given explicit expression.

A formal solution for the stationary distribution of queue-length at a fixed-cycle traffic light is found for a fairly general distribution of arrivals and for a single stream of vehicles which either all turn left or else all go straight on or turn right. (We assume that the vehicles are driving on the right of the road.) Some inequalities are derived for the expected queue-length and for the expected delay per vehicle.

Investigations in the Partially Balanced Incomplete Block (PBIB) designs having three or more associate classes have been limited to the works of Vartak [14], Raghavarao [7], Roy [9], Singh and Shukla [12] and Tharthare [13]. In this article we study the combinatorial properties, construction and non-existence of the cubic designs exhibiting a three associate class association scheme. The method of construction, discussed in this article, gives a new way of arranging a $p^3$ factorial experiment in blocks of sizes different from $p$ and $p^2$.

Majinder [3] obtained an upper bound for the number of disjoint blocks in BIB designs. In this paper we give an upper bound for the number of disjoint blocks in (i) Semi-regular GD designs, (ii) PBIB designs with two associate classes having triangular association scheme, (iii) PBIB designs with two associate classes having $L_2$ association scheme, and (iv) PBIB designs with three associate classes having rectangular association scheme. The main tools used to establish the results of this paper are the theorems proved by (i) Bose and Connor [1], (ii) Raghavarao [4] and (iii) Vartak [6].

Consider a finite projective space $PG(r - 1, s)$ of $r - 1$ dimensions, $r \geqq 3$, based on the Galois field $GF_s$, where $s = p^h, p$ being a prime. A set of distinct points in $PG(r - 1, s)$ is said to be a non-collinear set, if no three are collinear. The maximum number of points in such a non-collinear set is denoted by $m_3(r, s)$. It is the object of this paper to find a new upper bound for $m_3(r, s)$. This bound is of importance in the theory of factorial designs and error correcting codes. The exact value of $m_3(r, s)$ is known when either $r \leqq 4$ or when $s = 2$. When $r \geqq 5, s > 3$, the best values for the upper bound on $m_3(r, s)$ are due to Tallini [10] and Barlotti [1]. Our bound improves these when $s = 3$ or when $s$ is even.