This paper deals with the domains of attraction of the stable distributions and the normalizing coefficients associated with distributions in those domains of attraction. Using the notation $F \varepsilon \mathscr{D}(\alpha)$ and $F \varepsilon \mathscr{D}_\mathscr{N}(\alpha)$ to mean that the distribution function $F$ is in the domain of attraction and the domain of normal attraction respectively of a stable law of characteristic exponent $\alpha$, the following result is obtained: if $F \varepsilon \mathscr{D}(\alpha)$ and $G \varepsilon \mathscr{D}(\beta)$, where $0 < \alpha \leqq \beta \leqq 2$, and if $\{B_n\}$ and $\{C_n\}$ are normalizing coefficients respectively of $F$ and $G$, then $F \ast G \varepsilon \mathscr{D}(\alpha)$ and its normalizing coefficients are $\{(B^\alpha_n + C^\alpha_n)^{1/\alpha}\}$. Two more specialized results are obtained on convolutions of distribution functions in $\mathscr{D}(2)$, namely: (i) if $F \varepsilon \mathscr{D}_\mathscr{N}(2)$ and $G \varepsilon \mathscr{D}(2)\backslash\mathscr{D}_\mathscr{N}(2)$, then $F \ast G\varepsilon \mathscr{D}(2)\backslash\mathscr{D}_\mathscr{N}(2)$, and (ii) if $F$ and $G$ are distribution functions, and if the four tail probabilities vary regularly with exponent $-2$ and involve possibly four different slowly varying functions, then $F, G$ and $F\ast G$ are in $\mathscr{D}(2)$. These latter two results hold only for $\mathscr{D}(2)$ and not for $\mathscr{D}(\alpha)$ for $0 < \alpha < 2$, thus adding two exceptional properties to the normal law within the family of stable laws.

We consider repeated independent sampling from one member of an exponential family of probability distributions. The probability that the sample mean of $n$ such observations falls into some set $S_n$ is, by definition, a "large deviations", "small deviations", or "medium deviations" problem depending on the location of the set $S_n$ relative to the expectation of the distribution. We present a theorem which allows the accurate approximation of all such probabilities under a wide variety of circumstances. These approximations are shown to yield simple and numerically accurate expressions for the small sample power functions of hypothesis tests in the exponential family. Various large sample properties of exponential families are presented, many of which are seen to be extensions and refinements of familiar large deviations results. The method employed is to replace the given exponential family by a suitably modified normal translation family, which is shown to approximate the original family uniformly well over any bounded subset of the parameter space. The simple and tractable nature of normal translation families then provides our results.

For a random vector $(X_1,\cdots, X_k)$ having a $k$-variate normal distribution with zero mean values, Slepian [16] has proved that the probability $P\{X_1 < c_1,\cdots, X_k < c_k\}$ is a non-decreasing function of correlations. The present paper deals with the "two-sided" analogue of this problem, namely, if also the probability $P\{|X_1| < c_1,\cdots, |X_k| < c_k\}$ is a non-decreasing function of correlations. It is shown that this is true in the important special case where the correlations are of the form $\lambda_i\lambda_j\rho_{ij}, \{\rho_{ij}\}$ being some fixed correlation matrix (Section 1), and that it is true locally in the case of equicorrelated variables (Section 3). However, some counterexamples are offered showing that a complete analogue of Slepian's result does not hold in general (Section 4). Some applications of the main positive result are mentioned briefly (Section 2).

Kiefer and Wolfowitz (1959) proved that the optimal design for estimating the highest coefficient in polynomial regression is supported by certain Tchebycheff points. Hoel and Levine (1964) showed that the optimal designs for extrapolation in polynomial regression were all supported by the Tchebycheff points. These results were extended by Kiefer and Wolfowitz (1965) to cover nonpolynomial regression problems involving Tchebycheff systems and the large class of designs supported by the Tchebycheff points was characterized. In the present paper it is shown that the optimal design for estimating any specific parameter is supported by one of two sets of points for Tchebycheff systems with certain symmetry properties. Different proofs of the Kiefer-Wolfowitz results are also presented. The author wishes to thank Professor Kiefer for providing one of the counterexamples in Section 6.

The asymptotic chi-square distribution of the log-likelihood ratio is used to obtain approximate confidence intervals for the reliability of any system which may be represented by a monotone function of Bernoulli variates. This generalizes the results of A. Madansky, Technometrics, November 1965, for series, parallel and series-parallel systems. The method used is to parameterize the log-likelihood equation so as to find the interval of parameter values which keeps the log-likelihood less than or equal to the specified quantile of the chi-square distribution. This is done by introducing an operator depending upon the parameter, a fixed point of which is the solution of the likelihood ratio equation, and by showing the operator is a contractive map and hence has a unique fixed point depending continuously on the parameter. The solution can be found simply by iteration.

Suppose that $X_1, X_2, \ldots, X_n$ are random variables uniformly distributed over the simplex of points $x_1, x_2, \cdots, x_n$ such that $x_i \geqq 0$ for $i = 1, 2, \cdots, n$ and $\sum^n_1 x_i \leqq 1$. The distribution of $X = \sum^n_1 c_iX_i$ for constants $c_i$ satisfying $c_1 > c_2 > \cdots > c_n > 0$ is easily seen to be given by \begin{equation*}\begin{align*}\tag{1.1}P(X \leqq x) &= 1 - \sum^r_{j = 1} (c_j - x)^n\lbrack c_j\prod_{i \neq j} (c_j - c_i)\rbrack^{-1} \\ &= x^n\lbrack\prod^n_{i = 1} c_i\rbrack^{-1} - \sum^n_{j = r + 1} (x - c_j)^n\lbrack c_j\prod_{i \neq j} (c_i - c_j)\rbrack^ {-1}\end{align*}\end{equation*} for $0 \leqq x \leqq c_1$, where $r$ is the largest positive integer such that $x \leqq c_r$. A geometric derivation is given in Section 2 which relies on a principle of inclusion and exclusion and identifies the terms in (1.1) as volumes of various simplices. It is remarked in Section 3 that the argument of Section 2 extends in principle to give the joint distribution of a set of different linear combinations of $X_1, X_2, \cdots, X_n$. Finally, the relations to the theory of order statistics from a uniform distribution and to the theory of serial correlation are noted. A simple derivation of (1.1) will now be sketched. Suppose that $Z_1, Z_2, \cdots, Z_{n + 1}$ are $n + 1$ independent random variables each with the standard exponential density function $\exp (-z)$ for $0 \leqq z < \infty$. By a well-known argument involving the partial fraction expansion of the characteristic function (cf, Box (1954) Theorem 2.4), the density of $Z = \sum^n_1 c_jZ_j$ is found to be $\sum^n_1 (w_j/c_j) \exp (-z/c_j)$ where \begin{equation*}\tag{1.2}w_j = c^{n-1}_j\lbrack\prod_{i \neq j} (c_j - c_i)\rbrack^{-1}.\end{equation*} Now a set of random variables $X_1, X_2, \cdots X_n$ uniformly distributed over the simplex as described above may be represented as $X_j = Z_j/Y$ for $j = 1, 2, \cdots, n$ where $Y = \sum^{n+1}_1 Z_j$, and the set $X_1, X_2, \cdots, X_n$ thus created is distributed independently of $Y$. Thus $X = \sum^n_1 c_jX_j$ may be represented as \begin{equation*}\tag{1.3}X = Z/Y\end{equation*} where $X$ and $Y$ are independent. In a relation like (1.3) the marginal distributions of $Z$ and $Y$ together with the fact of independence of $X$ and $Y$ are sufficient to determine uniquely the distribution of $X$. To see this, note that $\operatorname{cf} (\log X)\cdot\operatorname{cf} (\log Y) = \operatorname{cf} (\log Z)$, where $\operatorname{cf}$ denotes the characteristic function. Thus, the characteristic function of $\log X$ is determined by those of $\log Y$ and $\log Z$. Now if $Z$ were known to have the scaled exponential density $c^{-1} \exp (-z/c)$ for $z \geqq 0$ while $Y$ had the gamma density $y^n \exp (-y)/n!$ for $y \geqq 0$, it would follow that $X$ had the scaled beta density $nc^{-1}(1 - x/c)^{n-1}$ for $0 \leqq x \leqq c$. But the density of $Z$ is, in fact, a mixture of scaled exponentials. Thus, the density of $X$ is the same mixture of scaled betas and is given by $\sum^n_1 w_jf_j(x)$, where the $w_j$'s are defined by (1.2) and \begin{equation*}\begin{align*}\tag{1.4}f_j(x) &= nc^{-1}_j (1 - x/c_j)^{n-1}\quad\text{for} 0 \leqq x \leqq c_j \\ &= 0\quad\text{otherwise}\end{align*}.\end{equation*} The result just given yields immediately the first line of (1.1). To derive the second line of (1.1), set $X_{n + 1} = 1 - \sum^n_1 X_j$ and note that $c_1 - X = c_1 X_{n + 1} + (c_1 - c_n)X_n + \cdots + (c_1 - c_2)X_2$ has a distribution of the same type as $X$. Applying the first line of (1.1) to $c_1 - X$ yields the second line of (1.1). The distribution of $\sum^{n + 1}_1d_iX_i$ is also given by (1.1), where $X_1, X_2, \cdots, X_{n + 1}$ are uniformly distributed over the simplex $x_i \geqq 0$ for $i = 1, 2, \cdots, n + 1$ and $\sum^{n + 1}_1 x_i = 1$, and where constants $d_i$ satisfy the condition \begin{equation*}\tag{1.5}d_1 > d_2 > \cdots > d_r \geqq x > d_{r + 1} > \cdots > d_{n + 1}.\end{equation*} To see this, let $c_i = d_i - d_{n + 1}$ and notice that $\sum^{n + 1}_1 d_iX_i = X + d_{n + 1}$.

Assume that $x_1, x_2, \cdots$ are independent random variables with expectation $\mu > 0$ and finite variance $\sigma^2$. Let $s_k = x_1 + \cdots + x_k (k = 1, 2, \cdots)$. For any family of positive, non-decreasing, eventually concave functions $f_c$ defined on the positive real numbers and indexed by $c > 0$ such that $f_c \rightarrow \infty$ as $c \rightarrow \infty$, define \begin{equation*}\begin{align*}\tau &= \tau(c) = \text{first}\quad k \geqq 1\quad\text{such that}\quad s_k > f_c(k) \\ &= \infty\quad\text{if no such}\quad k\quad\text{exists} \end{align*}.\end{equation*} The stopping variable (sv) $\tau$ arises in various problems in probability and statistics. For example, the sequential statistical procedures of [2], [3], and [8] involve sv's similar to $\tau$ (see also the next to last paragraph of this section). Suppose that the family $\{f_c:c > 0\}$ is such that we may define $\lambda = \lambda(c)$ by \begin{equation*}\tag{1}\mu\lambda = f_c(\lambda). \end{equation*} In [9] it is shown under conditions on the joint distribution of $x_1, x_2, \cdots$ weaker than the above that \begin{equation*}\tag{2}E\tau \sim\lambda\quad(c \rightarrow \infty)\end{equation*} for a certain class of families $\{f_c\}$. In Section 2 of this note it is shown that if $f_c(x) = cx^\alpha$ for some $0 \leqq \alpha < 1$, then $\tau$ (suitably normalized) is asymptotically normally distributed whenever $(s_n - n\mu)/\sigma n^{\frac{1}{2}}$ is. (See also remarks (a) and (b) in Section 4.) This extends Heyde's result [5], [7], valid when the $x_k$ have a common distribution and $\alpha = 0$. Assume now for simplicity that $f_c(x) \equiv c$ and $x_1, x_2, \cdots$ have a common distribution. If $x_1 \geqq 0$ the random variable $M(c)(N(c))$ defined by \begin{equation*}\tag{3}M(c) = \sup\{n:s_n \leqq c\} (N(c) = \sum^\infty_{n = 1}I\{s_n \leqq c\})\end{equation*} is of interest in renewal theory; and the observation that $M(c) = N(c) = \tau(c) - 1$ allows one to study $M(N)$ by studying the stopping variable $\tau$. In the general case it has been noted by several authors that the behavior of $\tau$ and $N$ may differ in important respects. (For example, if $E(x_1^-)^2 = \infty$, it is known that $EN = \infty$ [6], whereas it is easy to show that $E\tau < \infty, E\tau \sim c/\mu$ (e.g. [9]).) In Section 3 we point out that some knowledge of $M(N)$ can be obtained in a direct fashion from relevant knowledge of $\tau$. Our methods throughout involve finding convenient estimates for the probability that $s_k$ crosses the curve $f_c$ for the first time at some index $k < n$ and then falls back below the curve at time $n$.

Let $x'(t) = (x_1(t),x_2(t)),\quad(t = 1, 2, \cdots)$ be a two dimensional, Gaussian, vector process. Let the process $x'(t)$ have the representation \begin{equation*}\tag{1.1}x'(t) = \sum^p_{m = 0}B_my (t - m),\end{equation*} where \begin{equation*}\begin{align*}\tag{1.2} B_m &= \{b_{ijm}; i,j = 1, 2\}; \\ y'(t) &= (y_1(t), y_2(t)); \\ y_l(t) &= \sigma_l(t)\epsilon_l(t)\quad (l = 1, 2)\end{align*}.\end{equation*} The random variables $\epsilon_l(t)$ are independently and normally distributed with mean zero and variance unity. $p$ is a finite positive integer. The coefficients $B_m = (b_{ijm})_{2 \times 2}$ are finite real constants, and $\sigma^2_l(t)$ are non-random sequence of positive numbers which are not, in general equal, but do satisfy the conditions \begin{equation*}\tag{1.3}N^{-1}\sum^N_{t = 1}\sigma^2_l(t) = \nu_l < \infty\quad (\text{as} N \rightarrow \infty),\end{equation*} and $L \leqq \sigma^2_l(t) \leqq U < \infty\quad (t = 1, 2, \cdots).$ The relation (1.1) is a multivariate representation of a finite moving average process with time trending coefficients. Consider the matrix \begin{equation*}\begin{align*}\tag{1.4}F (\lambda) &= \begin{pmatrix}f_{11}(\lambda) & f_{12}(\lambda) \\ f_{21}(\lambda) & f_{22}(\lambda)\end{pmatrix} \\ &= G(\lambda)\begin{pmatrix}\nu_1 & 0 \\ 0 & \nu_2\end{pmatrix} G^{\ast'}(\lambda)\end{align*},\end{equation*} where $G(\lambda) = \sum^p_{m = 0} B_me^{im\lambda}$ and $G^\ast(\lambda)$ is its complex conjugate. Under the condition (1.3), Herbst [1] has defined $f_{11}(\lambda)$ and $f_{22}(\lambda)$ as the spectral densities of the processes $x_1(t)$ and $x_2(t)$ respectively, and considered their estimation. Here we generalize Herbst [1] results to a vector process and show that under the conditions (1.3) and (3.3) $f_{12}(\lambda)$, which is defined as the cross spectral density of the process $x_1(t)$ and $x_2(t)$, can consistently be estimated.

The discrete Student's $t$ distribution is discussed. In particular, we find that all finite odd order moments are zero, although the distribution need not be symmetric. A numerical comparison with the symmetric binomial shows the higher kurtosis of the Student form.

This paper presents some examples of "multidimensional incomplete block designs," a class of designs introduced as such by Potthoff [2] to provide analogues for the familiar two-factor incomplete block design. Much of the content of [2] was subsequently presented in [3], [4], [5], and [6]. Three-dimensional incomplete block ("3DIB") designs and four-dimensional incomplete block ("4DIB") designs for additive models are discussed in general terms in [3] and [4] respectively; our purpose in this paper will be to provide examples to which the tools developed in [3] and [4] can be applied. The original results of this paper have been presented in [1].

Let there be two samples $X_1, X_2, \cdots, X_m$ and $Y_1, Y_2, \cdots, Y_n(N = m + n)$ from two populations with continuous cdf's $F(x)$ and $G(y)$. Let the first $i$ ordered observations (out of $N$ combined observations) contain $m_ix$'s and $n_iy$'s $(m_i + n_i = i)$ where $m_i$ and $n_i$ are random numbers. To test \begin{equation*}\tag{0.1}H_0:F = G\end{equation*} against alternative that they are different we propose the statistic \begin{equation*}\tag{0.2}S^{(N)}_r = \sum^r_{i = 1} a_iz_i + (m - m_r)(N - r)^{-1}(\sum^N_{r+1}a_i) - \frac{1}{2}(m + n)\end{equation*} based on the first $r$ ordered observations only where $a^N_i = a_i = \sum^N_{j=N-i+1} 1/j,$ and \begin{align}z_i &= 1, \text{if the} i\text{th ordered observation is an} x_i, \\ &= 0, otherwise.\end{align} The statistic is the asymptotically most powerful rank test for censored data under the Lehmann alternative and is equivalent to the Savage statistic [14] when $r = N$. It is also known to maximize the minimum power over IFRA (or IFR) distributions asymptotically. Exact and large sample properties of $S^{(N)}_r$ are studied and a $k$-sample extension of it is also considered. Various tables are also provided to facilitate the use of the $S^{(N)}_r$ statistic.

Consider random samples from two independent distributions with absolutely continuous distribution functions $F(z)$ and $F(z - \theta)$, respectively. For testing the hypotheses $\theta = 0$ against $\theta > 0$, Hodges and Lehmann [6] show that the Pitman asymptotic efficiency of $W$, the Mann-Whitney form of the Wilcoxon statistic, with respect to the $t$-statistic is never smaller than .864 and in their 4th Berkeley Symposium paper [7] they indicate this efficiency is almost always greater than or equal to 1 for distributions with tails at least as heavy as those of a normal distribution. Hence for distributions with heavier tails, $W$ is a more robust statistic than the $t$-statistic. For the moment, consider a single sample of size $n$ from a distribution with absolutely continuous distribution function $F(z - \theta)$. For distributions with heavier tails some authors have proposed the $\alpha$-trimmed mean as an estimate of $\theta$; that is, the mean based on the middle $n - 2\lbrack n\alpha\rbrack$ observations. Tukey [16] and Huber [8] study this statistic when $F$ is a contaminated normal distribution. Bickel [3] studies the asymptotic relative efficiency properties of the $\alpha$-trimmed mean relative to the mean for the class of continuous distributions with symmetric, unimodal densities. For estimating the location of a Cauchy distribution, Rothenberg, Fisher and Tilanus [12] show the trimmed mean based on the middle 24 percent of the observations is the most efficient trimmed mean relative to the maximum likelihood estimate. This single sample statistic suggests a Mann-Whitney statistic based on trimmed samples. It is hoped that the effects of contamination by gross errors in the underlying distributions can be considerably reduced by considering such a statistic. We have the benefits of using a simple and well known rank statistic and, at the same time, of being able to increase the efficiency by adjusting the trimming proportions according to the weight in the tails of the underlying distributions. This type of statistic is also related to rank tests for censored data which have been studied by Basu [2], Gastwirth [5] and Sobel [14], [15]. If we have samples size $m$ and $n$ from absolutely continuous distributions corresponding to $F(z)$ and $F(z - \theta)$, respectively, we denote by $W_\alpha$ the Mann-Whitney statistic based on the middle $m - 2\lbrack m\alpha\rbrack$ and $n - 2\lbrack n\alpha\rbrack$ observations of the samples. We refer to $W_\alpha$ as the $\alpha$-trimmed Mann-Whitney statistic. It is the purpose of this paper to investigate the Pitman asymptotic relative efficiency properties of $W_\alpha$ for a sub-class of absolutely continuous distributions. First some definitions are given in Section 2. In Section 3 and Section 4 we establish the asymptotic normality of $W_\alpha$, derive the efficiency of $W_\alpha$ relative to $W$ and give some examples. The results of Section 5 include a greatest lower bound on this efficiency. It is interesting to note that this bound is the same one found by Bickel [3] for the efficiency of the trimmed mean relative to the mean.

The first identity (Section 2) expresses a moment of a product of multivariate $t$ densities as an integral of dimension one less than the number of factors. This identity is applied to inference concerning the location parameters of a multivariate normal distribution. The second identity (Section 3) expresses the density of a linear combination of independently distributed multivariate $t$ vectors, a multivariate Behrens-Fisher density (Cornish (1965)), as an integral of dimension one less than the number of summands. The two-summand version of the second identity is essentially equivalent to the two-factor version of the first identity. A synthetic representation is given for the random vector, generalizing Ruben's (1960) representation in the univariate case. The second identity is applied to multivariate Behrens-Fisher problems. The third identity (Section 4), due to Picard (Appell and Kampe de Feriet (1926)), expresses the moments of Savage's (1966) generalization of the Dirichlet distribution as a one-dimensional integral. A generalization of Picard's identity is given. Picard's identity is applied to inference about multinomial cell probabilities, to components-of-variance problems, and to inference from a likelihood function under a Student $t$ distribution of errors.

In [1], Anderson and Goodman investigated statistical inference about Markov chains for random samples of independently, identically distributed chains, in [6] Goodman obtained results for single chains and in [3] Baum and Petrie studied inferences for probabilistic functions of Markov chains. Most of their results are either for a large number of identical chains or for very long chains. In this paper, we shall be mainly concerned with the distributions of certain probabilistic functions of a finite chain and the admissibility of these functions as test statistics. Some similar and related results in this respect, especially those for a two state Markov chain may be found in the works of many authors, such as Fisz [5], chapter 11, Goodman [6], Lehmann [8], pp. 115-6, and Mood [9]. The results in Goodman [6] dealt more generally with $s$ state Markov chain with $s \geqq 2$ and certain extensions and supplementary results can be found in [2] and [7]. Throughout this paper, we shall assume that our Markov chain is defined on a finite state space and has a discrete time parameter which takes on nonnegative integers. Our derivations make use of a vector representation of an arbitrary Markov chain as given in Section 2 below. In Section 3, a special sequence of probabilistic functions of a Markov chain is defined and shown to be independently, identically distributed multinomial random vectors under certain assumptions about the transition probabilities. The admissibility of test statistics based on some of these functions is shown in Section 4 and a few examples are given in Section 5.

When testing $p$-variate distributions for a shift in location, two important nonparametric competitors of Hotelling's $T{}^2$ are the multivariate extensions $W$ of the Wilcoxon test and $M$ of the normal score test. Bounds on their asymptotic relative efficiency (ARE) have been investigated by Hodges-Lehmann [6] and Chernoff-Savage [4] in the univariate case and by Bickel [3] and Bhattacharyya [1] in the multivariate case. The univariate normal score test has the commendable property that for all continuous distributions, its ARE with respect to the $t$-test exceeds 1 and with respect to the Wilcoxon test it exceeds $\pi/6$. This naturally raises the question of whether or not the multivariate extension $M$ inherits this property and if not, what the lower bounds on its ARE with respect to $W$ and $T{}^2$ are. In this paper, we answer this question by providing an example where the ARE of $M$ with respect to both $W$ and $T{}^2$ is arbitrarily close to zero for some direction. The example consists of a gross error distribution which places most of its mass on a hyperplane and has marginals with high sixth moments. Bickel [3] mentioned a similar property of the ARE of $W$ with respect to $T{}^2$. His proof for the case $p = 2$ is, however, incorrect. We show that for the type of gross error model considered by Bickel, the above ARE is bounded strictly away from zero. We correct his proof by constructing a distribution which also places high mass on a line but is not of the gross error type.

A Chernoff and Savage theorem on the asymptotic normality of 2-sample linear rank statistics is here established for random sample sizes. The proof parallels that of Pyke and Shorack (1968), hereafter referred to as PS. A mild restriction on the underlying distributions is needed in the present situation. A result of Pyke (1968) on the weak convergence of the 1-sample empirical process for random sample sizes in the ordinary uniform metric is here extended to other metrics. This extension provides an essential step in the present proof and is also of separate interest. The results extend immediately to $c$-samples.

The zonal polynomials of the positive definite real symmetric matrices, which appear in the expansions of the functions occurring in many multivariate non null distributions and moment formulae are eigenfunctions of the Laplace Beltrami operator. The resulting differential equation gives a recurrence relation between the coefficients from which they can be calculated.

We prove the following THEOREM. Let $\{S_n, n \geqq 1\}$ be a martingale, $S_0 = 0, X_n = S_n - S_{n-1}, \gamma_{\nu n} = E(|X_n|^\nu)$ and $\beta_{\nu n} = (1/n) \sum^n_{j = 1} \gamma_{\nu j}$. Then for all $\nu\geqq 2$ and $n = 1, 2, \cdots$ \begin{equation*}\tag{1.1}E(|S_n|^\nu) \leqq C_\nu n^{\nu/2}\beta_{\nu n},\end{equation*} where \begin{equation*}\tag{1.2}C_\nu = \lbrack 8(\nu - 1) \max (1, 2^{\nu - 3})\rbrack^\nu.\end{equation*} As shown by Chung ([3], pp. 348-349) an inequality of Marcinkiewicz and Zygmund ([5], p. 87) implies that the theorem holds (possibly with a different value of $C_\nu$) whenever the $X$'s are independent. In the same way the above theorem is implied by the generalization of the Marcinkiewicz-Zygmund result given by Burkholder ([2], Theorem 9). However, our proof is elementary. Although our choice of $C_\nu$ is not the best possible, it is explicit. For the case of independent $X$'s, von Bahr ([6], p. 817) has given a bound for $E(|S_n|^\nu)$ which may sometimes involve powers of $\beta_{\nu n}$ higher than 1. Finally Doob ([4], Chapter V, Section 7) has treated the case when the $X$'s form a Markov chain. After proving some lemmata in Section 2, we give the proof of the theorem in Section 3. The case of exchangeable random variables is dealt with in Section 4.

The usual technique of deriving the asymptotic normality of a quantile of a sample in which the random variables are all independent and identically distributed [cf. Cramer (1946), pp. 367-369] fails to provide the same result for an $m$-dependent (and possibly non-stationary) process, where the successive observations are not independent and the (marginal) distributions are not necessarily all identical. For this reason, the derivation of the asymptotic normality is approached here indirectly. It is shown that under certain mild restrictions, the asymptotic almost sure equivalence of the standardized forms of a sample quantile and the empirical distribution function at the corresponding population quantile, studied by Bahadur (1966) [see also Kiefer (1967)] for a stationary independent process, extends to an $m$-dependent process, not necessarily stationary. Conclusions about the asymptotic normality of sample quantiles then follow by utilizing this equivalence in conjunction with the asymptotic normality of the empirical distribution function under suitable restrictions. For this purpose, the results of Hoeffding (1963) and Hoeffding and Robbins (1948) are extensively used. Useful applications of the derived results are also indicated.

This work is an investigation of a nonparametric approach to the problem of testing for a shift in the level of a process occurring at an unknown time point when a fixed number of observations are drawn consecutively in time. We observe successively the independent random variables $X_1, X_2, \cdots, X_N$ which are distributed according to the continuous cdf $F_i, i = 1, 2, \cdots, N$. An upward shift in the level shall be interpreted to mean that the random variables after the change are stochastically larger than those before. Two versions of the testing problem are studied. The first deals with the case when the initial process level is known and the second when it is unknown. In the first case, we make the simplifying assumption that the distributions $F_i$ are symmetric before the shift and introduce the known initial level by saying that the point of symmetry $\gamma_0$ is known. Without loss of generality, we set $\gamma_0 = 0$. Defining a class of cdf's $\mathscr{F}_0 = \{F:F$ continuous, $F$ symmetric about origin$\}$, the problem of detecting an upward shift becomes that of testing the null hypothesis $H_0:F_0 = F_1 = \cdots = F_N,\quad\text{some}\quad F_0 \varepsilon\mathscr{F}_0,$ against the alternative $H_1:F_0 = F_1 = \cdots = F_m > F_{m + 1} = \cdots = F_N,\quad\text{some}\quad F_0 \varepsilon\mathscr{F}_0$ where $m(0 \leqq m \leqq N - 1)$ is unknown and the notation $F_m > F_{m + 1}$ indicates that $X_{m + 1}$ is stochastically larger than $X_m$. For the second situation with unknown initial level, the problem becomes that of testing the null hypothesis $H_0^\ast:F_1 = \cdots = F_N$, against the alternatives $H_1^\ast: F_1 = \cdots = F_m > F_{m + 1} = \cdots = F_N$, where $m(1 \leqq m \leqq N - 1)$ is unknown. Here the distributions are not assumed to be symmetric. The testing problem in the case of known initial level has been considered by Page [11], Chernoff and Zacks [2] and Kander and Zacks [7]. Assuming that the observations are initially from a symmetric distribution with known mean $\gamma_0$, Page proposes a test based on the variables $\operatorname{sgn} (X_i - \gamma_0)$. Chernoff and Zacks assume that the $F_i$ are normal cdf's with constant known variance and they derive a test for shift in the mean through a Bayesian argument. Their approach is extended to the one parameter exponential family of distributions by Kander and Zacks. Except for the test based on signs, all the previous work lies within the framework of a parametric statistics. The second formulation of the testing problem, the case of unknown initial level, has not been treated in detail. The only test proposed thus far is the one derived by Chenoff and Zacks for normal distributions with constant known variance. In both problems, our approach generally is to find optimal invariant tests for certain local shift alternatives and then to examine their properties. Our optimality criterion is the maximization of local average power where the average is over the space of the nuisance parameter $m$ with respect to an arbitrary weighting $\{q_i, i = 1, 2, \cdots, N: q_i \geqq 0, \sum^N_{i = 1} q_i = 1\}$. From the Bayesian viewpoint, $q_i$ may be interpreted as the prior probability that $X_i$ is the first shifted variate. Invariant tests with maximum local average power are derived for the case of known initial level in Section 2 and for the case of unknown initial level in Section 3. In both cases, the tests are distribution-free and they are unbiased for general classes of shift alternatives. They all depend upon the weight function $\{q_i\}$. With uniform weights, certain tests in Section 3 reduce to the standard tests for trend while a degenerate weight function leads to the usual two sample tests. In Section 4, we obtain the asymptotic distributions of the test statistics under local translation alternatives and investigate their Pitman efficiencies. Some small sample powers for normal alternatives are given in Section 5.

Consider, for the moment, a balanced fixed two way layout of the analysis of variance assuming interactions and replications. Suppose a hypothesis of interest is that the row effects are all equal. A test statistic for such a hypothesis is the ratio of the mean square for rows (MSR) divided by the mean square for error (MSE). Another test procedure frequently used in practice is as follows: Test whether the interaction effects are zero. If it is decided that these effects are not zero, then test for the equality of row effects by MSR/MSE. If it is decided that the interaction effects are zero, then test for the equality of row effects by MSR divided by the pooled mean square error. That is, the pooled mean square error consists of the sum of squares for error plus the sum of squares for interaction divided by the sum of degrees of freedom for error and interaction. This latter type of procedure is called a "sometimes pooling" procedure. For a more general description and discussion of such procedures see Bozivich, Bancroft, and Hartley (1956). In this note we consider the general linear hypothesis model. We prove, in this general framework, that the "sometimes pooling" procedure is an admissible test procedure. The proof follows from a well known invariance result and a theorem of Matthes and Truax (1967). The "sometimes pooling" procedure can be viewed as a test procedure which depends on the outcome of a preliminary test. It is interesting to note that estimation procedures which depend on a preliminary test were found to be inadmissible for the squared error loss function. (See Cohen (1965).) In the next section we state the model and prove the admissibility result.