We consider a probabilistic method for constructing certain transition functions which have a given conservative matrix as their initial derivative matrix. The technique originated in a conjecture of Reuter (see [9]). Kingman (see [5]) has considered similar problems but no proof of the original conjecture has appeared. The methods used are intended to confirm a remark made by Chung (see [1] page 158) to the effect that it should be possible to view sticky (regular) boundary points as a suitable limiting case of non-sticky (non-regular) boundary points. The probabilistic construction in Reuter's conjecture can be used to motivate some general analytical constructions of transition functions. These constructions and the connection with modern boundary theory are discussed in the last two sections.

Asymptotic behavior of a class of confidence regions, based on rank statistics, for the regression parameter vector is considered. These regions are shown to be asymptotically bounded and ellipsoidic in probability. Asymptotic normality of their center of gravities is also proved. It is noted that the asymptotic efficiencies of these regions when defined in terms of ratio of Lebesgue measures corresponds to that of corresponding test statistics that are used to define these regions. Similar conclusion holds for their center of gravities, where now asymptotic efficiency is defined as inverse ratio of their generalized limiting variances. Also a class of consistent estimators is given for some functionals of the underlying distributions. Finally simultaneous confidence intervals, based on the above center of gravity, for linear parametric functions are shown to have asymptotic coverage probability $1 - \alpha$. Basic to this work are two papers, one by the author [4] and one by Jureckova [3].

In this paper are studied asymptotically minimax properties of nonparametric tests. It is shown that for two different definitions of asymptotically minimax test, sequences of Wilcoxon tests are asymptotically minimax. The first definition concerns asymptotic power of tests with fixed level, and the other is a more decision-theoretic one.

The importance of the theory of Latin squares and mutually orthogonal Latin squares is due to the fact that many well-known experimental designs and/or combinatorial systems are either equivalent or can be derived from Latin squares or sets of mutually orthogonal Latin squares; viz., balanced incomplete block designs, partially balanced incomplete block designs, group divisible designs, $F$-square designs, lattice designs, balanced weighing designs, orthogonal arrays, Hadamard matrices, affine planes, projective planes, partial geometries, nets, error correcting codes, strongly regular graphs and many more designs and combinatorial systems (see [7] for more details). The problem of determining whether or not a set of $r$ mutually orthogonal Latin squares of order $n$, say an $O(n, r)$ set, can be embedded in a larger set of $t$ mutually orthogonal Latin squares of order $n$, is one of the many important problems in the theory of mutually orthogonal Latin squares. An obvious and interesting problem in the area of embedding is to find a necessary and sufficient condition under which an $O(n, r)$ set can be embedded in an $O(n, t)$ set for $1 \leqq r < t \leqq n - 1$, where by an $O(n, 1)$ set we mean a set consisting of a single Latin square of order $n$. Some of the principal published results on embedding are discussed below. Mann [9] proved that if a Latin square $L$ is of order $n = 4t + 2$ (or alternatively $4t + 1$) with a subsquare of order $2t + 1$ (or $2t$) in which all entries are from a set of $2t + 1$ (or $2t$) numbers except for possibly $t$ (or $\lbrack t - \frac{1}{2}\rbrack$) or less of the cells, then $L$ cannot be embedded in an $O(n, 2)$ set. Thus, Mann's embedding conditions depend upon the value of $n$ and the combinatorial structure of $L$. Parker [11] proved that if the $O(n, r)$ set $S$ contains a sub $O(r + 1, r)$ set, then $S$ can be embedded in an $O(n, r + 1)$ set only if $n = (r + 1)^2$ or $n \geqq (r + 1)(r + 2)$. These embedding conditions depend upon $r, n$, and the combinatorial structure of the set $S$. Shrikhande [15] proved that any $O(n, n - 3)$ set $S$ can be embedded in an $O(n, n - 1)$ set for all $n > 4$. His result does not require any knowledge about the combinatorial structure of the set $S$. Bruck [2], who generalized Shrikhande's [15] results to some extent, utilized net theory to show that any $O(n, n - 1 - d)$ set $S$ can be embedded in an $O(n, n - 1)$ set regardless of the combinatorial structure of $S$ provided that $n > (d - 1)(d^3 - d^2 + d + 2)/2$. Utilizing the above results and some group theory, some new results have been found for embedding $O(n, r)$ sets in $O(n, t)$ sets, $r < t$. The results are presented in the form of several theorems, propositions and corollaries. The first proposition is concerned with a necessary and sufficient condition for embedding $O(n, 2)$ sets in $O(n, 3)$ sets. Theorem 3.1 gives a sufficient condition for embedding an $O(n, 1)$ set in an $O(n, \lambda)$ set, $2 \leqq \lambda \leqq$ smallest prime in the prime power decomposition of $n$, less one. Theorem 3.2 states the conditions under which the method of Theorem 3.1 fails for embedding certain $O(n, 1)$ sets in a larger set. The next theorem states the conditions under which certain $O(n, 1)$ sets cannot be embedded in a larger set. Some results on properties of inverses of Latin squares and on products have also been presented. Additional results in this paper are embodied in Theorem 3.4 and relate to the enumeration of Latin squares of order $n$ for those Latin squares which have orthogonal mates and for those which do not have an orthogonal mate. Theorem 3.4 gives a lower bound for $n$ even (odd) on the number of Latin squares of degree one (at least degree two). All Latin squares of order 2, 3, 4, 5, 6 and 7 are classified with respect to degree of orthogonality.

Let $\mathbf{X}_1: p \times n$ and $\mathbf{X}_2: p \times n$ be real random variables having the joint density function \begin{equation*}\tag{1.1} (2\pi)^{-pn}|\mathbf{\Sigma}_0|^{-\frac{1}{2}n} \exp \{-\frac{1}{2} \operatorname{tr}\mathbf{\Sigma}_0^{-1}(\mathbf{X} - \mathbf{\nu})(\mathbf{X} - \mathbf{\nu})'\},\quad - \infty \leqq \mathbf{X} \leqq \infty\end{equation*} where \begin{equation*}\mathbf{X} = \binom{\mathbf{X}_1}{\mathbf{X}_2},\quad \mathbf{\Sigma}_o = \begin{pmatrix}\mathbf{\Sigma}_1 & -\mathbf{\Sigma}_2 \\ \mathbf{\Sigma}_2 & \mathbf{\Sigma}_1\end{pmatrix},\quad \nu = \begin{pmatrix}\mathbf{\mu}_1 & -\mathbf{\mu}_2 \\ \mathbf{\mu}_2 & \mathbf{\mu}_1\end{pmatrix} \binom{\mathbf{M}_1}{\mathbf{M}_2},\end{equation*} $\mathbf{\Sigma}_1: p \times p$ is a real symmetric positive definite (pd) matrix, $\mathbf{\Sigma}_2: p \times p$ is a real skew-symmetric matrix, $\mathbf{\mu}_j: p \times q$ and $\mathbf{M}_j: q \times n (j = 1, 2)$, are given matrices or their joint density does not contain $\mathbf{\Sigma}_1, \mathbf{\Sigma}_2, \mathbf{\mu}_1, \mathbf{\mu}_2$ as parameters. Then it has been shown by Goodman [10] that the distribution of the complex matrix $\mathbf{Z} = \mathbf{X}_1 + i\mathbf{X}_2, (i = (-1)^{\frac{1}{2}})$, is complex Gaussian and its density function is given by \begin{equation*}\tag{1.2} N_c(\mathbf{\mu}\mathbf{M}, \mathbf{\Sigma}) = \pi^{-pn}|\Sigma|^{-n} \exp \{ -\operatorname{tr} \mathbf{\Sigma}^{-1} (\mathbf{Z} - \mathbf{\mu M})(\mathbf{\bar{Z}} - \mathbf{\overline{\mu M}})'\}\end{equation*} where $\mathbf{\Sigma} = \mathbf{\Sigma}_1 + i\mathbf{\Sigma}_2$ is Hermitian pd, i.e. $\mathbf{\bar{\Sigma}}' = \mathbf{\Sigma, \mu} = \mathbf{\mu}_1 + i\mathbf{\mu}_2$ and $\mathbf{M} = \mathbf{M}_1 + i\mathbf{M}_2$. Goodman [5], Wooding [17], James [6], Al-Ani [1], and Khatri [8], [9], [10], [11] have studied distributions derived from a sample of a complex $p$-variate normal distribution. Some important concepts and necessary notation are given below. \begin{align*} \tilde\Gamma_m(a) &= \pi^{\frac{1}{2}m(m - 1)} \prod^m_{i = 1} \Gamma(a - i + 1), \\ \lbrack a \rbrack_\kappa &= \prod^m_{i = 1} (a - i + 1)_{k_i} = \tilde\Gamma_m(a, k)/\tilde\Gamma_m(a)\end{align*} where $\kappa = (k_1, k_2, \cdots, k_p)$ is a partition of the integer $k$ and $\tilde\Gamma_m(a, \kappa) = \pi^{\frac{1}{2}m(m - 1)} \prod^m_{i = 1} \Gamma(a + k_i - i + 1).$ The hypergeometric functions are defined as $_p\tilde{F}_q(a_1, \cdots, a_p; b_1, \cdots, b_q; \mathbf{A, B}) = \sum^\infty_{k = 0} \sum_k \frac{\prod^p_{i = 1} \lbrack a_i \rbrack_\kappa \tilde{C}_\kappa (A) \tilde{C}_\kappa (\mathbf{B})}{\prod^q_{i = 1} \lbrack b_i \rbrack_\kappa \tilde{C}_\kappa (\mathbf{I}_m)k!}$ or when $\mathbf{B} = \mathbf{I}_m$ we denote it by $_p\tilde{F}_q(a_1, \cdots, a_p; b_1, \cdots, b_q; \mathbf{A})$ and $\tilde{C}_\kappa(\mathbf{A})$ is a zonal polynomial of a Hermitian matrix $\mathbf{A}$ and is given as a symmetric function of the characteristic roots of $\mathbf{A}$. The non-central distributions of the characteristic roots concerning the classical problems of the equality of two covariance matrices, MANOVA model, and canonical correlation coefficients have been found by James [6] and Khatri [8], [10]. Here for the three cases mentioned, we give the general moment and the density which is expressed in terms of Meijer's $G$-function [13], [14], for $W^{(p)} = \prod^p_{i = 1} (1 - w_i)$, where the $w_i, i = 1, 2, \cdots, p$ are the characteristic roots in the above cases. The moments and densities are analogous to those given in the real case [7], [15]. Further the density functions of $U$ and Pillai's $V$ criteria in the complex central case are obtained for $p = 2$, and from the non-central complex multivariate $F$ distribution various independence relationships are shown and independent beta variables are obtained.

The sample canonical correlations between two sets of variates are given a representation as the roots of a determinantal equation involving independent matrix variates having simple standardized distributions. This result is applied to obtain asymptotic formulas for the non-null distributions of three criteria for testing the hypothesis of independence of two sets of variates.

We study a two-sided procedure proposed by E. S. Page for detecting a change in the location of the distribution of a sequence of independent observations which are ordered in time. We approximate the null distribution of Page's statistic and the power of his test for finite sequences. When the procedure is applied to an infinite sequence we approximate the average run length. In order to obtain these approximations we find the distribution function of the range of a Wiener process with drift and the Laplace transform of the time at which the range first exceeds some given value.

Let $Z = \{z_1, z_2, \cdots, z_m\}$ be a $p \times m$ random matrix where $z_i$ are independently distributed according to $p$-variate normal distributions with means $\mu_i$ and common covariance matrix $\Lambda = (\lambda_{ij})(> 0$, positive definite) and let $nS_n = n(s_{ij})$ be a $p \times p$ matrix which is independent of $Z$ and is subject to a central Wishart distribution $W_p(\Lambda, n)$ with $n$ degrees of freedom and covariance matrix $\Lambda$. Hotelling's generalized $T_o^2$-statistic is then defined by \begin{equation*}\tag{1.1} T_o^2 = \operatorname{tr} S_n^{-1}ZZ' = \sum^m_{i=1} z_i'S_n^{-1}z_i\end{equation*} which is the statistic proposed by Hotelling [4], [5] for testing the hypothesis $H: M = \{\mu_1, \cdots, \mu_m\} = 0$ against $K: M \neq 0$. The distribution of $T_0^2$ when $H$ is true has been treated by several authors: for example, Hotelling [5], Ito [6], Siotani [13], [14], Pillai and Samson [12], and Davis [3]. Even in the null case, the exact distribution of $T_0^2$ is not available except for certain special values of $p$ and $m$. When $m = 1, T_0^2$ reduces to Hotelling's generalization of "Student's" $t$ and if $p = 1, (1/m)T_0^2$ is simply $F$-statistic. Hence the non-null distributions in these cases are known exactly. For the general non-null case, Constantine [2] has obtained the exact distribution as well as moments of $T_0^2/n$ using the generalized Laguerre polynomials of matrix argument. Unfortunately this distribution is valid only over the range $0 \leqq T_0^2/n < n 1$ and hence not so useful since we are usually interested in the upper tail of the distribution. Siotani [15] has treated an asymptotic expansion for the non-null distribution of $T_0^2$ according to the basic idea due to Welch [17] and James [9]. The same problem has been attacked by Ito [7], using the integral representation of the characteristic function of $T_0^2$. However formulas of these authors are inadequate for a good approximation to the distribution since they have only the terms up to order $n^{-1}$ (for some numerical information, see [8]) and also somewhat inconvenient terms for numerical work. Khatri and Pillai have evaluated the moments of their statistic $U^{(p)}$ (a constant times $T_0^2$) and given approximate distributions of $U^{(p)}$ (and hence of $T_0^2$) in the light of the first four general noncentral moments, the summary of which can be obtained in their recent paper [10]. In this paper an asymptotic expansion for the non-null distribution of $T_0^2$ is given up to the terms of order $n^{-2}$, in which the effect of the noncentrality is contained in the form $s_j = \operatorname{tr} \Omega^j, j = 1, 2, \cdots$ where $\Omega = \Lambda^{-1}MM'$. This is carried out by expanding the characteristic function of $T_0^2$ along the Welch-James idea and by inverting the resultant into the corresponding expansion of the distribution function or the density function.

Matching problems are discussed in many elementary probability books, such as Feller (1968). In one version of the problem, as described by Hodges and Lehmann (1964), the photographs of $n$ film stars are paired randomly with $n$ photographs of the same stars taken when they were babies, and the distribution of the number of correct matches is derived. In this paper we shall study the same problem when the photographs are paired on the basis of various measurements that are made on them, rather than randomly. For example, suppose that $r$ different facial measurements are made on the photograph of each star and that $s$ facial measurements are made on each baby photograph. By comparing these measurements, it will typically be possible to devise a method for pairing the photographs that will yield a larger number of correct matches than would be obtained from random pairing. In fact, the procedures that will be developed in this paper can be regarded as formalizations of the heuristic procedures that a person follows when he pairs the photographs on the basis of perceived resemblances. In other versions of the same problem, dental records of parents are to be matched with dental records of their children, or measurements made on the chest X-rays of $n$ individuals are to be matched with other medical records of these same individuals. The problems described here are related in principle to problems of document linkage that have been treated in the statistical literature [see, e.g., DuBois (1969) and the references given there] but the models and methods that are used here seem to be new and unrelated to the models and methods that have previously been used in such problems. For any positive integer $k$, we shall let $R^k$ denote the space of all $k$-dimensional vectors $z = (z_1, \cdots, z_k)$, where $- \infty < z_i < \infty$ for $i = 1, \cdots, k$. Now let $T$ denote an $r$-dimensional random vector $(r \geqq 1)$, let $U$ denote an $s$-dimensional random vector $(s \geqq 1)$, and suppose that $T$ and $U$ have some specified joint distribution over the space $R^{r + s}$. We shall assume that a random sample of $n$ vectors $(t_1, u_1), \cdots, (t_n, u_n)$ has been drawn from this joint distribution. It is assumed, however, that before the values in this sample can be observed, each vector $(t_i, u_i)$ in the sample is broken into two separate vectors, namely the vector $t_i$ with $r$ components and the vector $u_i$ with $s$ components. The vectors $t_1, \cdots, t_n$ are then observed in some random order, say $\nu_1, \cdots, \nu_n$ and the vectors $u_1, \cdots, u_n$ are observed in some independent random order, say $w_1, \cdots, w_n$. As a result of this randomization, it is not known how the vectors $\nu_1, \cdots, \nu_n$ and the vectors $w_1, \cdots, w_n$ were paired in the original sample. It is assumed that a priori (i.e., before the specific values of $\nu_1, \cdots, \nu_n$ and $w_1, \cdots, w_n$ are observed) all $n!$ ways of pairing $\nu_1, \cdots, \nu_n$ with $w_1, \cdots, w_n$ are equally likely to reproduce the original sample. The observed vectors $\nu_1, \cdots, \nu_n$ and $w_1, \cdots, w_n$ will be called the values of a broken random sample from the specified joint distribution of $T$ and $U$. The general problem to be considered here is that of pairing the observed vectors $\nu_1, \cdots, \nu_n$ with the observed vectors $w_1, \cdots, w_n$ in order to reproduce as many of the vectors $(t_i, u_i)$ from the original sample as possible. The application of this model to the problem of matching the photographs of $n$ individuals with their baby photographs or matching two sets of medical records of $n$ individuals should be clear. The important assumption that we have made here is that the observations for the $n$ given individuals can be regarded as the values in a random sample of size $n$ from some larger population of individuals for which the probability distribution is known. In this paper we shall assume that the joint distribution of $T$ and $U$ can be represented by a joint pdf $f$ of the following form: \begin{equation*}\tag{1.1} f(t, u) = \alpha(t)\beta(u) e^{\gamma(t)\delta(u)} \quad t \in R^t, u \in R^s,\end{equation*} where $\alpha, \beta, \gamma, \text{and} \delta$ are arbitrary real-valued functions of the indicated vectors. If either $r = 1$ or $s = 1$, and if the joint distribution of $T$ and $U$ is a multivariate normal distribution, then their joint pdf will be of the form (1.1). A multivariate normal distribution of this type is undoubtedly the most important special case of (1.1). In particular, if both $t$ and $u$ are one-dimensional, the pdf of every bivariate normal distriubtions is of the form (1.1). Another example of a vivariate pdf of the form (1.1) is \begin{align*} f(t, u) &= t e^{-t(1+u)} \\ t > 0, u > 0, \\ &= 0 \\ \text{otherwise}\end{align*} We shall now present a summary of the specific problems that will be considered in this paper and some of the results that will be obtained. In Section 2, the problem of pairing the vectors $\nu_1, \cdots, \nu_n$ with the vectors $w_1, \cdots, w_n$ in order to maximize the probability of a completely correct set of $n$ matches is considered. It is shown that the probability is maximized if the values of $\gamma(\nu_1), \cdots, \gamma(\nu_n)$ are ordered from smallest to largest, the values of $\delta(w_1), \cdots \delta(w_n)$ are similarly ordered, and corresponding terms in these two orderings are paried with each other. This solution is also the maximum likelihood solution for the problem of pairing $\nu_1, \cdots, \nu_n$ with $w_1, \cdots, w_n$. In Section 3 this maximum likelihood solution is applied to the multivariate normal distribution and is shown to have a natrual and intuitive interpretation in terms of regression. In Section 4, we consider the problem of choosing a vector $w_j$ from the set $w_1, \cdots, w_n$ in order to maximize the probability of correctly matching one specified vector $\nu_1$ from the set $\nu_1, \cdots, \nu_n$. It is shown that if $\gamma(\nu_i)$ is the minimum or the maximum of the $n$ values $\gamma(\nu_1), \cdots, \gamma(\nu_n)$, then $\nu_i$ should be paired with a vector $w_j$ for which $\delta(w_j)$ is a minimum or a maximum, respectively. For intermediate values of $\gamma(\nu_i)$, the solution is shown to be more complicated. In Section 5, the problem of pairing $\nu_1, \cdots, \nu_n$ with the vectors $w_1, \cdots, w_n$ in order to maximize the expected number of correct matches is considered. Although the general solution of this problem is complicated, it is shown here again that the vector $\nu_i$ for which $\gamma(\nu_i)$ is a minimum should always be paired with the vector $w_j$ for which $\delta(w_j)$ is a minimum and the vector $\nu_i$ for which $\gamma(\nu_i)$ is a maximum should always be paired with the vector $w_j$ for which $\delta(w_j)$ is a maximum. In particular, it follows that when $n = 3$, the solution to this problem and the maximum likelihood solution are always identical. In Section 6, sufficient conditions are given under which, for an arbitrary value of $n$, the maximum likelihood solution will also maximize the expected number of correct matches. The simplest and most striking sufficient condition given there, but also the most severe condition, is that $\lbrack \max_i \gamma(\nu_i) - \min_i\gamma(\nu_i) \rbrack \lbrack \max_j \delta(w_j) - \min_j \delta(w_j) \rbrack \leqq 1.$ Finally, in Section 7, some examples are given in which these sufficient conditions are not satisfied and the maximum likelihood solution does not maximize the expected number of correct matches.

Tests of marginal homogeneity in a two-way contingency table given by [1], [3], and [13] do not seem to lend themselves easily to extension to the problem of $m$-way marginal homogeneity in an $N$-way $r \times r \times \cdots \times r$ contingency table, $m < N$. The principle of minimum discrimination information estimation and the associated minimum discrimination information statistic applied in [5] to the problem of marginal homogeneity in an $r \times r$ contingency table can be easily extended to the case of a multidimensional contingency table. Estimates of the cell entries under the hypotheses of $m$-way marginal homogeneity are given. Relationships among the tests of homogeneity for $m$-way, $m = 1, 2, \cdots, N - 1$, marginals are given by an analysis of information. Numerical results are given for two sample $3 \times 3 \times 3$ tables, and two $5 \times 5$ tables.

A law of the iterated logarithm is derived for the empirical distribution functions of a sequence of independent identically distributed random variables. Convergence is in the uniform topology on the space of functions on the reals with discontinuities of the first kind only. The proof depends on a law of the iterated logarithm for independent identically distributed vector-valued random variables.

Studden (1968) showed that the optimal design for estimating any specific regression coefficient or parameter is supported by one of two sets of points for Tchebycheff systems with certain symmetry properties. In this paper we consider a Tchebycheffian Spline Regression Function, defined on an interval, and show that the optimal design for estimating any specified regression coefficient is supported on the same set of points. Familiarity with the notation and terminology used in the paper of Studden referred to above is assumed.

Hajek (1962) has obtained asymptotically most powerful rank order tests for simple linear regression with non-stochastic predictors. His findings are extended here to the multiple linear regression model with stochastic predictors, including the situations where the predictors are partially informed. The proposed tests are shown to be conditionally distribution-free. Their asymptotic properties and efficiencies are studied, and the asymptotic optimality is established under the conditions of Wald (1943).

For a bivariate population, with a probability density $p(|x - \theta|)$ where $p(t)$ is strictly decreasing for $t \geqq 0$, when a single observation is taken, the usual confidence sets are circles of constant radius centered at the observed value. Previous results of Kiefer imply that these circles have the minimax property of minimizing the maximum expected Lebesgue measure of the confidence sets for a given lower confidence level. It is now shown here that subject to mild conditions on $p(t)$, the confidence circles are also essentially unique in having the minimax property. The result generalizes the result proved previously for the bivariate normal case.

Consideration is given to minimum variance unbiased estimation when the choice of estimators is restricted to a finite-dimensional linear space. The discussion gives generalizations and minor extensions of known results in linear model theory utilizing both the coordinate-free approach of Kruskal and the usual parametric representations. Included are (i) a restatement of a theorem on minimum variance unbiased estimation by Lehmann and Scheffe; (ii) a minor extension of a theorem by Zyskind on best linear unbiased estimation; (iii) a generalization of the covariance adjustment procedure described by Rao; (iv) a generalization of the normal equations; and (v) criteria for existence of minimum variance unbiased estimators by means of invariant subspaces. Illustrative examples are included.

Since the notion of completeness for a family of distributions was introduced by Lehmann and Scheffe [7], a problem of interest has been to determine conditions under which a complete sufficient statistic exists for a family of multivariate normal distributions. One approach to this problem, first formulated for a completely random model in some work by Graybill and Hultquist [2] and extended to a mixed linear model by Basson [1], has a basic assumption of commutativity for certain pairs of matrices. In the present paper some of the commutativity conditions and an associated eigenvalue condition assumed in the theorems on completeness in both [1] and [2] are replaced by the weaker requirement of a quadratic subspace. These subspaces, i.e., quadratic subspaces, are introduced and briefly investigated in Section 2 and are found to possess some rather interesting mathematical properties. The existence of $\bar\mathscr{A}$-best estimators (e.g., [12]) is also examined for several situations; and it is found that the usual estimators in the weighting factors for the recovery of interblock information in a balanced incomplete block design (treatments fixed and blocks random) have an optimal property when the number of treatments is equal to the number of blocks. Throughout the paper $(\mathscr{A}, (-,-))$ denotes the FDHS (finite-dimensional Hilbert space or finite-dimensional inner product space) of $n \times n$ real symmetric matrices with the trace inner product. The notation $Y \sim N_n(X\beta, \sum^m_{i=1} \nu_iV_i)$ means that $Y$ is an $n \times 1$ random vector distributed according to a multivariate normal distribution with expectation $X\beta$ and covariance matrix $\sum^m_{i=1} \nu_iV_i$; and for such a random vector the following is assumed: (a) $X$ is a known $n \times p$ matrix and $\beta$ is an unknown vector of parameters ranging over $\Omega_1 = R^p$. (b) Each $V_i(i = 1,2,\cdots, m)$ is a known $n \times n$ real symmetric matrix, $V_m = I$, and $\nu = (\nu_1,\cdots, \nu_m)'$ is a vector of unknown parameters ranging over a subset $\Omega_2$ of $R^m$. (c) The set $\Omega_2$ contains a non-void open set in $R^m$ and $\sum^m_{i=1} \nu_iV_i$ is a positive definite matrix for each $\nu \in \Omega_2$. (d) The parameters $\nu$ and $\beta$ are functionally independent so that the entire parameter space is $\Omega = \Omega_1 \times \Omega_2$. For the special case when $X = 0$ the notation $Y \sim N_n(0, \sum^m_{i=1} \nu_iV_i)$ is used and for this situation the parameter space $\Omega$ reduces to $\Omega_2$. The notation and terminology in the following sections is generally consistent with the usage in [12]. The adjoint of a linear operator $\mathbf{T}$ is denoted by $\mathbf{T}^\ast$ and the transpose of a matrix $A$ is denoted by A'. Additionally, the unique Moore-Penrose generalised inverse of a matrix $A$ is denoted by $A^+$, and as in [12], only real finite-dimensional linear spaces are considered.

The characteristic roots of the information matrix $(M)_T$ of a balanced $2^m$ fractional factorial design $T$ are obtained, when the parameters to be estimated include the general mean $\mu$, the main effect $A_i$, and the two-factor interaction $A_iA_j$ (briefly, $A_{ij}$), the remaining effects being assumed negligible. (If $(M)_T$ is nonsingular, $T$ is a design of resolution $V$.) It is well known that $T$ depends on five nonnegative integers $(\mu_0, \mu_1, \mu_2, \mu_3, \mu_4)$, called its "index set." In Srivastava (1970), the special case when $\mu_0 = \mu_4$ and $\mu_1 = \mu_3$ was considered; in this paper, the theory is presented for the general case. As a by-product of this work, we obtain a class of useful necessary conditions on the set $(\mu_0, \mu_1, \mu_2, \mu_3, \mu_4)$ such that a design $T$ with this index set may (combinatorially) exist. If $(M)_T$ is nonsingular, and $(V)_T = \lbrack(M)_T\rbrack^{-1}$, an explicit expression (as a function of the $\mu_i$) has been obtained for $\operatorname{tr}(V)_T$; similar expressions for $|(V)_T|$ and $\operatorname{ch}_{\max}(V)_T$ can be easily written down using our results. One reason why $\operatorname{tr}(V)_T$ (rather than the other two criteria) should be used for comparing balanced resolution $V$ fractions is given. Finally, it is shown (through an example of a previously unknown design with resolution $V m = 7$) that for a given $N$ (the number of runs), an (existing) optimal balanced design (optimal with respect to, say, the trace criterion) does not necessarily satisfy the restriction $(\mu_0 = \mu_4$ and $\mu_1 = \mu_3$), and may be distinct from the design which is optimal in the restricted class. (Scores of other such examples may be found in Srivastava and Chopra (1970a), where the results of this paper are used in a basic manner.) Thus the need for considering designs with general index sets (which is accomplished in the present paper) becomes obvious.

Necessary and sufficient conditions are given for a triangular array of numbers to be expectations of order statistics of some nonnegative random variable. Using well-known recurrence relations, the expectations of all order statistics of the largest sample size, $n$, in the triangular array, or the expectations of the smallest of every sample size up to and including $n$ are sufficient to determine the whole array. The former are reduced to a Stieltjes moment problem, the latter to a Hausdorff moment problem. These results are applied to show that for every sample size, there is a positive random variable with geometrically increasing expectations of order statistics with arbitrary ratio and expectation of smallest order statistic. However, only the degenerate distributions have geometrically increasing expectations of order statistics for more than one sample size, even when the ratio and mean of the smallest order statistic can depend on the sample size. These results were required for a study of participation in discussion groups.

This paper deals with the finite part of multiple autoregressive series. The conditions of stationarity are derived and the inverse of the covariance matrix is evaluated (without assumption of stationarity).

Let $\mathbf{x}: p \times 1$ be distributed $N(\mathbf{\mu}, \mathbf{\Sigma})$ where $\mathbf{\mu}$ and $\mathbf{\Sigma}$ are both unknown. Let $\mathbf{S}$ be the sum of product matrix of a sample of size $N$. To test the hypothesis of sphericity, namely, $H_0:\mathbf{\Sigma} = \sigma^2\mathbf{I}_p$, where $\sigma^2 > 0$ is unknown, against $H_1:\mathbf{\Sigma} \neq \sigma^2\mathbf{I}_p$, Mauchly [10] obtained the likelihood ratio test criterion for $H_0$ in the form $W = |\mathbf{S}|/\lbrack(\operatorname{tr} \mathbf{S})/p\rbrack^p$. Thus the criterion $W$ is a power of the ratio of the geometric mean and the arithmetic mean of the roots $\theta_1, \theta_2, \cdots, \theta_p$ of $|\mathbf{S} - \theta\mathbf{I}| = \mathbf{0}$ (see Anderson [1]). In the null case, Machly [10] gave the density of $W$ for $p = 2$ and Consul [3], [4] for any $p$ in terms of Meijer's $G$-function defined in the next section. In this paper we have obtained the general moments of $W$ both in real and complex cases for arbitrary covariance matrices, and also the corresponding distributions of $W$ in terms of the $G$-function.

In this paper we characterize the multivariate normal distribution through the mean vector of the distribution. In Theorem 3, we find that the multivariate normal distribution can also be characterized through its variance, even though unknown, provided the variance-covariance matrix is free of the parameter involved. We are able to determine the mean vector to within two unknowns, the variance-covariance matrix and a constant vector.

A conjecture of Hanurav (1968), that subject only to a mild restriction, any arbitrary estimate is admissible in the class of all unbiased estimates of its expectation (which is a population function) is shown to be false.