It is now widely recognized that from the point of view of robustness, nonparametric tests, such as the Wilcoxon or normal scores test, should be used in practice for the two-sample location problem instead of the classical $t$-test. But until recently there were no robust estimates for the difference between the two location parameters. In a recent paper [4] Hodges and Lehmann proposed a solution for this problem. After this paper it is clear that all the arguments that can be used in favor of the Wilcoxon test as against the classical $t$-test can be used in favor of the estimate $\operatorname{med} (Y_j - X_i)$ for the difference between the locations as against the classical estimate $(\bar Y - \bar X)$. In their paper quoted above, Hodges and Lehmann propose a whole class of nonparametric estimates corresponding to a class of nonparametric tests both for the two-sample and the one-sample location problems. To indicate this correspondence suppose $h(X, Y)$ is a test-statistic, nonparametric or otherwise for the equality of the locations of $X$ and $Y$. After having observed $(X, Y)$ we estimate the difference between the two location parameters by the amount of shift required to match the samples $X$ and $Y$ in such a way that $h(X, Y)$ is close to its expected value when the shift is zero. For a more precise definition of the estimates the reader is referred to (1.2) below. For a corresponding definition of the one-sample estimates the reader is referred to (3.3). Since the difference between the two one-sample estimates for location is an estimate for the difference between the two location parameters in the two-sample problem the paper of Hodges and Lehmann throws open a whole class of estimates for location in the two-sample problem. The aim of this paper is two-fold. First, how do these estimates compare among themselves in the Behrens-Fisher situation where the scale parameters of the populations can possibly differ? Second, how do these estimates compare with the classical estimate when the scales differ? A basic requirement to be able to answer the above questions is the asymptotic normality of the estimates in the Behrens-Fisher situation and this is shown under fairly general conditions in Sections 2 and 3. In answer to the first question the following main result is proved: If the $\Psi^\ast$-score test is the best linear rank order test when the underlying distribution is $F^\ast$ then the difference between the two one-sample estimates based on the $\Psi^\ast$-score test is more efficient than the simultaneous two-sample estimate based on the $\Psi^\ast$-score test. It follows as a particular case that the difference between the two one-sample estimates based on the normal scores test is more efficient than the simultaneous two-sample estimate based on the normal scores test when the prototype distribution is normal. In answer to the second question, the following result is shown. The differences between the one-sample Hodges-Lehmann estimates has all the advantages over the classical estimate in the case of inequality of variances as in the case of equality of variances. For a more precise statement the reader is referred to the end of Section 4.

Let $(X^{(1)}_{j1}, \cdots, X^{(p)}_{j1}), \cdots, (X^{(1)}_{jn_j}, \cdots, X^{(p)}_{jn_j})$ be random samples of size $n_j$ from the $c$ continuous $p$-variate distribution functions $F_j(\mathbf{x}) = F(\mathbf{x} - \mathbf{\theta}_j)$ where $\mathbf{x} = (x_1, \cdots, x_p), \mathbf{\theta}_j = (\theta_{ij}, \cdots, \theta_{pj})$ and $j = 1, \cdots, c$. This paper is concerned with the problem of testing the hypothesis $H : \mathbf{\theta}_1 = \mathbf{\theta}_2 = \cdots = \mathbf{\theta}_c$ against the alternative that all $\mathbf{\theta}_j$ are not equal on the basis of the above $c$ samples. When we especially consider the asymptotic efficiency of test, the following alternative $K$ is adopted, $K : \mathbf{\theta}_j = \mathbf{\theta} + \mathbf{\nu}_j/N^{\frac{1}{2}},\quad\mathbf{\nu}_j = (\nu_{ij}, \cdots, \nu_{pj}),\quad j = 1, \cdots, c.$ We shall develop in this paper some test procedures for the hypothesis $H$ which are originated from the paper of Chernoff-Savage [2]. When $c = 2$, that is multivariate two-sample tests, Sugiura [7] has proposed a test statistic of Wilcoxon type. On the other hand, Puri [5] has derived univariate several-sample tests in general type including the test of Kruskal-Wallis [4]. It will be shown that the latter are corresponding to the case $p = 1$ in this paper and the former is a particular one among the case $c = 2$ of our tests.

The power and efficiency of robust procedures have been considered for parametric alternatives by Tukey (1946), Hoeffding (1951), Lehmann (1953), (1959), Chernoff and Savage (1958), Capon (1961), and others. For instance, it has been shown that the two-sample normal scores statistic is optimal for normal translation alternatives in that for these alternatives, it is locally most powerful [12], [15], [4] in the class of all rank statistics and it is asymptotically efficient [6], [4] in the class of all statistics. However, for other types of translation alternatives, the normal scores statistic does not have these optimal properties. In this paper, optimality properties for non-parametric classes of alternatives are treated. In particular, statistics that maximize the minimum power asymptotically over classes of non-parametric alternatives will be considered. If one takes (1-power) as the risk function, these statistics are asymptotically minimax. It turns out that in this sense the Wilcoxon statistic is asymptotically minimax over a class of one-sided alternatives $(F, G)$ with $\rho(F, G) \geqq \Delta$, where $\rho$ is the Kolmogorov distance, while the normal scores statistic is asymptotically minimax for a non-parametric class of translation type alternatives. For the problem of estimating differences in location, the normal scores estimate of Hodges and Lehmann (1963) is shown to be minimax when the risk function is asymptotic variance. Finally, the results are used to obtain asymptotic efficiencies that are defined for non-parametric classes of alternatives.

Non-orthogonal partitions of $n \times n$ Latin squares into $(n + 1)$ or $(n - 1)$ groups may be useful when further treatments are to be added to experimental designs in Latin squares. The different methods of constructing these partitions for $n = 4, 5$ or 6 are considered here. It is found that a partition into $(n + 1)$ groups is always possible when the Latin square has a directrix and a partition into $(n - 1)$ groups is possible whether or not there is a directrix. A complete enumeration of all possible partitions is given for the partition of $4 \times 4$ squares into 3 or 5 groups and that of $5 \times 5$ squares into 6 groups. Examples only are given for the partition of the $5 \times 5$ squares into 4 groups and the partition of the $6 \times 6$ squares into either 5 or 7 groups.

Let $I$ be the closed unit interval, with the usual topology; II the set of probabilities on $I$, with the weak$^\ast$ topology: $\mu_n \rightarrow \mu$ in II if and only if $\int_I f d\mu_n \rightarrow \int_I f d\mu$ for each continuous, real-valued $f$ on $I$. For $\mu, \nu$ in II, recall that $\mu \equiv \nu$ means: $\mu(A) = 0$ if and only if $\nu(A) = 0$ for all Borel subsets $A$ of $I$. Of course, $\equiv$ is an equivalence relation. The graph of $\equiv$, namely the set of $(\mu, \nu) \varepsilon II \times II$ with $\mu \equiv \nu$, is Borel (2.11 of [2]). One result of this paper is: there is no Borel selector for $\equiv$; that is, there is no Borel subset of II meeting each $\equiv$ class in exactly one point. Let $\Sigma(\equiv)$ be the $\sigma$-field of all Borel subsets of II saturated under $\equiv$, that is, containing with $\mu$ all $\nu \equiv \mu$. If there were a Borel selector for $\equiv$, there would be a Borel function $f$ on II with $f(\mu) = f(\nu)$ if and only if $\mu \equiv \nu$, and $\Sigma(\equiv)$ would be separable. However, (1) Proposition. $\Sigma(\equiv)$ is inseparable. The proof of (1) is based on the following idea of Blackwell. Let $\mathscr{F}$ be a $\sigma$-field, and $P$ a probability on $\mathscr{F}$. Say $P$ is continuous if each $\mathscr{F}$-atom has outer $P$-measure 0, and say $P$ is $0 - 1$ if $P(A) = 0$ or 1 for all $A \varepsilon \mathscr{F}$. (2) Lemma (Blackwell). If $P$ is continuous and $0 - 1, \mathscr{F}$ is inseparable. Thus, (1) follows from (3) Theorem. There is a continuous $0 - 1$ probability on $\Sigma(\equiv)$. Two proofs of (3) will be given in Sections 2 and 4 respectively. Section 3 contains a result on random distribution functions [3], which may be of independent interest, and which is used in Section 4. Section 5 deals with the coarser equivalence relation $\approx$, where $\mu \approx \nu$ means: $\mu(A) = 0$ if and only if $\nu(A) = 0$ for all open subsets $A$ of $I$. Now $\approx$ is induced by a Borel function (3.5 of [2]). More is true: (4) Theorem. There is a Borel selector for $\approx$.

This paper consists of three parts: first a new simple proof of the semimartingale theorem of Doob is given, next the limit function is identified as the derivative of a certain $\sigma$-additive set function. Finally it is shown how the approach of Sparre-Andersen and Jessen can be generalized to give the convergence and the identification of the limit function for a semimartingale.

Let $\{X_k: k = 0, 1, 2, \cdots\}$ be a Markov process with state space $(\mathscr{S}, \mathscr{B})$ and denote by $f$ a real-valued $\mathscr{B}$-measurable function on $\mathscr{S}$. In [1] and [6] it was shown that for a Markov process with a single ergodic class, if the transition probabilities $P(x, A) = P\lbrack X_1 \varepsilon A \mid X_0 = x\rbrack$ are stationary and satisfy Doeblin's condition ([2]), if the conditional moment generating function of $f, \int_{\mathscr{S}} e^{tf(y)} P(x, dy)$, satisfies a certain boundedness condition in $t$ uniformly in $x$ and if $\int_{\mathscr{S}}f(y)\pi(dy) = 0$ where $\pi$ is the unique stationary probability measure for the process, then an exponential bound exists for the convergence of $S_n = n^{-1}\lbrack f(X_1) + \cdots + f(X_n)\rbrack$ to zero which is independent of the initial probability measure. That is, for every $\epsilon > 0$ there exist positive constants $A$ and $\rho < 1$ such that for all initial measures $\nu$, $P_\nu\lbrack|S_n| \geqq \epsilon\rbrack \leqq A\rho^n\quad\text{for} n = 1, 2, \cdots.$ The purpose of this paper is to study the extent to which such a result holds when the Cesaro averages $n^{-1}\lbrack f(X_1) + \cdots + f(X_n)\rbrack$ are replaced by a more general class of averages $\sum^\infty_{k = 1} a_{n,k}f(X_k), n = 1, 2, \cdots$, where the infinite matrices $\{a_{n,k}\}$ satisfy the conditions (a) $\sum^\infty_{k = 1} |a_{n,k}| \leqq 1$ uniformly for $n = 1, 2, \cdots$; (b) $\lambda(n) = \max_k |a_{n,k}| \rightarrow 0$ as $n \rightarrow \infty$; (c) $\lim_{n \rightarrow \infty} \sum^\infty_{k = 1} a_{n,k} = 1$. Such matrices are called Toeplitz matrices (except that our Condition (b) is somewhat stronger than the one usually imposed, see e.g. [8]), and our concern will be with the rate at which the sequence of random variables $S_n = \sum^\infty_{k = 1} a_{n,k}f(X_k)$ tends to zero in probability as $n \rightarrow \infty$. We will obtain theorems analogous to those of [1] and [6] through a series of specializations of two basic theorems to be proved in Section 2. There, without the assumption of stationary transition functions, conditions are given under which for every $\epsilon > 0$, there exist positive constants $A$ and $\rho < 1$ such that for all initial measures $\nu$, \begin{equation*}\tag{0}P_\nu\lbrack|S_n| \geqq \epsilon\rbrack \leqq A\rho^{1/\lambda(n)},\quad n = 1, 2, \cdots.\end{equation*} Here $S_n$ is the Toeplitz average and $\lambda(n)$ is defined above in Condition (b). In Section 3, processes with stationary transition probabilities are studied and a condition on the conditional moment generating function of $f$ is introduced which is comparable to a condition introduced in [1] and which implies the first theorem of Section 2. In Section 4, Doeblin's condition is introduced and additional restrictions on the process and the Toeplitz matrices sufficient to yield (0) are considered. This study is a sequel to work begun in [3] and continued in [4] and [5]. For applications in which Toeplitz averages of stochastic processes are of interest and for further orientation on the subject of convergence rates for Toeplitz averages of random variables, the reader is referred to these papers.

A simplified solution is presented for the problem of finding a set of points and corresponding weights that will minimize the variance of the estimated value of a Chebyshev regression function at a point outside the interval of observations. This problem, among others, was solved by Kiefer and Wolfowitz [3] by means of game-theoretic methods. The solution here is based on a simple theorem in [2] and well known properties of Chebyshev systems of functions.

In this note, some recurrence relations are derived between the expected values of functions of order statistics from any arbitrary distribution with continuous cumulative distribution function (cdf). These recurrence relations are closely related to some results obtained by Srikantan [2].

Let $X_{i0}$ and $X_{ij} (i = 1, \cdots, n; j = 1, \cdots, k)$ be the independent measurements on the control and $j$th treatment in the $i$th block, with $P(X_{ij} \leqq x) = F_j(x - b_i)$ Here $b_i$ is the block $i$ nuisance parameter and the $F_j; j = 0, \cdots, k$, are assumed continuous. Nemenyi [5] suggests treatment-control comparisons based on the statistic $T = \max_j T_{0j}$ where $T_{0j}$ is defined by (2.1). It is shown here that, under the null hypothesis \begin{equation*}\tag{1.1}H_0:F_j = F \text{(unknown)},\quad j = 0, \cdots, k,\end{equation*} $T$ is neither distribution-free for finite $n$, nor asymptotically distribution-free. We also modify Nemenyi's procedure so that it is asymptotically distribution-free.

Agrawal (1964) and Shah (1965) have derived bounds for the number of common treatments between the blocks of PBIB designs. Agrawal's result is more general and is applicable to any $(v, b, r, k)$ design. In this note it is established that Agrawal's bounds are superior to those given by Shah.