The present investigation is concerned with the formulation of a multivariate approach for the construction of a class of aligned rank order tests for the analysis of variance (ANOVA) problem relating to two-way layouts. The problems of simultaneous testing and testing for ordered alternatives based on aligned rank order statistics are also considered. Various efficiency results pertaining to the proposed tests are studied. Let us consider a two factor experiment comprising $n$ blocks, each block containing $p(\geqq 2)$ plots receiving $p$ different treatments. In accordance with the two-way ANOVA model, we express the yield $X_{ij}$ of the plot receiving the $j$th treatment in the $i$th block as \begin{equation*}\tag{1.1}X_{ij} = \mu + \alpha_i + \tau_j + \epsilon_{ij}, \quad i = 1, \cdots, n, j = 1, \cdots p;\end{equation*} where $\mu$ stands for the mean effect, $\alpha_1, \cdots, \alpha_n$ for the block effects (may or may not be stochastic), $\tau_1, \cdots, \tau_p$ for the treatment effects (assumed to be non-stochastic), and $\epsilon_{ij}$'s are the residual error components. It is assumed that $\mathscr{\epsilon}_i = (\epsilon_{i1}, \cdots, \epsilon_{ip}), i = 1, \cdots, n$ are independent and identically distributed stochastic vectors having a continuous (joint) cumulative distribution function (cdf) $G(x_1, \cdots, x_p)$ which is symmetric in its $p$ arguments; (this includes the conventional situation of independence and identity of distributions of all the $np$ error components as a special case). We may set without any loss of generality $\sum^p_1 \tau_j = 0$, and frame the null hypothesis of no treatment effect as \begin{equation*}\tag{1.2}H_0 : \tau_1 = \cdots = \tau_p = 0.\end{equation*} The usual ANOVA test based on the variance-ratio criterion is valid only when $G$ is a $p$-variate (totally symmetric) multinormal cdf. For arbitrarily continuous cdf $G(x_1, \cdots, x_p)$, intra-block rank tests are due to Friedman [7], Brown and Mood [3], and Sen [21]; generalizations of these tests to incomplete layouts are due to Durbin [6], Benard and Elteren [1], and Bhapkar [2]. Hodges and Lehmann [9] have pointed out that intra-block rank tests do not utilize the information contained in the interblock comparisons, and hence, are comparatively less efficient. They have suggested the use of ranking after alignment and also considered the Wilcoxon's and Kruskal-Wallis tests based on aligned observations whose asymptotic efficiency have been studied very recently by Mehra and Sarangi [14]: (After the first draft of the paper was submitted, the author came to know through the editor about the paper by Mehra and Sarangi [14], submitted earlier to the Annals. In view of this, the overlapping part is not considered here.) The object of the present investigation is to formulate briefly the theory of rank order tests based on observations after alignment and through a multivariate approach to justify the validity and efficacy of the proposed tests (which include the earlier works as special cases).

Neyman and Scott [12] considered the problem of estimating the mean of a distribution after some fixed Gaussian inducing transformation had been applied to the observations. More specifically, if a random variable $x$ is observed then it is assumed that $\xi = f{}^{-1}(x)$ has a Gaussian distribution with mean $\mu$ and variance $\sigma^2$. They find the minimum variance unbiased estimator (MVUE), $\hat\theta$, of $\hat{\theta} = E(x)$ in terms of the MVUE's of $\mu$ and $\sigma^2$. The heart of their solution is in taking a Taylor series expansion of $f(\xi)$ around the origin and showing that the resulting infinite power series behaves like a finite one in the sense that the operations of taking expectations and summing can be reversed as required provided only that $f(\xi)$ is an entire function of second order or less. This paper exploits the Taylor series expansion of $f(\xi)$ around the mean to find the MVUE of $\phi^2 = E(x - \theta)^2$. It was motivated in part by the fact that recent research in cloud seeding has shown that the variance may be a more important parameter than the mean [8], [15], and also by a general concern with the class of recursive transformations defined on p. 651 of [12]. In addition the MVUE of Var $(\hat{\theta})$ for particular transformations is derived. The problems under consideration here have been considered by Finney [6], Sichel [14], and Meulenberg [11] for the logarithmic transformation i.e., $f(\xi) = e^{m\xi}$. Their results provide a check on mine. Schmetterer [13] has interpreted the results of Neyman and Scott in terms of the solution $h(\hat{\mu}, \hat{\sigma}^2)$ of the integral equation $E\lbrack h(\hat{\mu}, \hat{\sigma}^2)\rbrack = E(x)$ where $\hat{\mu}, \hat{\sigma}^2$ are the MVUE's for $\mu$ and $\sigma^2$. Kolmogorov [10] has also considered the problem of finding unbiased estimators in terms of the solutions of integral equations but he relies heavily upon the results of Blackwell [2] in using the sufficient statistics for $(\mu, \sigma^2)$ to turn unbiased but inefficient estimators into the MVUE. The problem discussed here can be formulated as an integral equation: viz., find $h\ast(\hat{\mu}, \hat{\sigma}^2)$ such that $E\lbrack h^\ast(\hat{\mu}, \hat{\sigma}^2_\rho)\rbrack = E\lbrack x - E(x)\rbrack^2$ but the present author has not attempted to solve this problem in this way. Rather he has approached the problem by a straightforward application of the method of Neyman and Scott.

Considerations on stochastic models frequently involve sums of dependent random variables (rv's). In many such cases, it is worthwhile to know if asymptotic normality holds. If so, inference might be put on a nonparametric basis, or the asymptotic properties of a test might become more easily evaluated for certain alternatives. Of particular interest, for example, is the question of when a weakly stationary sequence of rv's possesses the central limit property, by which is meant that the sum $\sum^n_1 X_i$, suitably normed, is asymptotically normal in distribution. The feeling of many experimenters that the normal approximation is valid in situations "where a stationary process has been observed during a time interval long compared to time lags for which correlation is appreciable" has been discussed by Grenander and Rosenblatt ([10]; 181). (See Section 5 for definitions of stationarity.) The general class of sequences $\{X_i\}_{-\infty}^\infty$ considered in this paper is that whose members satisfy the variance condition \begin{equation*}\tag{1.1}\operatorname{Var} (\sum^{a+n}_{a+1} X_i) \sim nA^2\text{uniformly in} a (n \rightarrow \infty) (A^2 > 0).\end{equation*} Included in this class are the weakly stationary sequences for which the covariances $r_j$ have convergent sum $\sum_1^\infty r_j$. A familiar example is a sequence of mutually orthogonal rv's having common mean and common variance. As a mathematical convenience, it shall be assumed (without loss of generality) that the sequences $\{X_i\}$ under consideration satisfy $E(X_i) \equiv 0$, for the sequences $\{X_i\}$ and $\{X_i - E(X_i)\}$ are interchangeable as far as concerns the question of asymptotic normality under the assumption (1.1). As a practical convenience, it shall be assumed for each sequence $\{X_i\}$ that the absolute central moments $E|X_i - E(X_i)|^\nu$ are bounded uniformly in $i$ for some $\nu > 2$ ($\nu$ may depend upon the sequence). When (1.1) holds, this is a mild additional restriction and a typical criterion for verifying a Lindeberg restriction ([15]; 295). We shall therefore confine attention to sequences $\{X_i\}$ which satisfy the following basic assumptions (A): \begin{equation*}\tag{A1}E(X_i) \equiv 0,\end{equation*}\begin{equation*}\tag{A2}E(T_a^2) \sim A^2 \text{uniformly in} a (n \rightarrow \infty) (A^2 > 0),\end{equation*}\begin{equation*}\tag{A3}E|X_i|^{2+\delta} \leqq M (\text{for some} \delta > 0 \text{and} M < \infty),\end{equation*} where $T_a$ denotes the normed sum $n^{-\frac{1}{2}} \sum^{a+n}_{a+1} X_i$. Note that the formulations of (A2) and (A3) presuppose (A1). We shall say, under assumptions (A), that a sequence $\{ X_i\}$ has the central limit property (clp), or that $T_1$ is asymptotically normal (with mean zero and variance $A^2$), if \begin{equation*}\tag{1.2}P\{(nA^2)^{-\frac{1}{2}}\sum^n_1 X_i \leqq z\} \rightarrow (2\pi)^{-\frac{1}{2}} \int^z_{-\infty} e^{-\frac{1}{2}t{}^2}dt\quad (n \rightarrow \infty).\end{equation*} The assumptions (A) do not in general suffice for (1.2) to hold. (The reader is referred to Grenander and Rosenblatt ([10]; 180) for examples in which (1.2) does not hold under assumptions (A), one case being a certain strictly stationary sequence of uncorrelated rv's, another case being a certain bounded sequence of uncorrelated rv's.) It is well known, however, that in the case of independent $X_i$'s the assumptions (A) suffice for (1.2) to hold. It is desirable to know in what ways the assumption of independence may be relaxed, retaining assumptions (A), without sacrificing (1.2). Investigators have weakened considerably the moment requirements (A2) and (A3) while retaining strong restrictions on the dependence. However, in many situations of practical interest, assumptions (A) hold but neither strong dependence restrictions nor strong stationarity restrictions seem to apply. Thus it is important to have theorems which take advantage of assumptions (A) when they hold, in order to utilize conclusion (1.2) without recourse to severe additional assumptions. A basic theorem in this regard is offered in Section 4. It is unfortunate that the additional assumptions required, while relatively mild, are not particularly amenable to verification, with present theory. This difficulty is alleviated somewhat by the strong intuitive appeal of the conditions. The variety of ways in which the assumption of independence may be relaxed in itself poses a problem. It is difficult to compare the results of sundry investigations in central limit theory because of the ad hoc nature of the suppositions made in each instance. In Section 2 we explore the relationships among certain alternative dependence restrictions, some introduced in the present paper and some already in the literature. Conditions involving the moments of sums $\sum^{a+n}_{a+1} X_i$ are treated in detail in Section 3. The central limit theorems available for sums of dependent rv's embrace diverse areas of application. The results of Bernstein [2] and Loeve [14], [15] have limited applicability within the class of sequences satisfying assumptions (A). A result that is apropos is one of Hoeffding and Robbins [11] for $m$-dependent sequences (defined in Section 2). In addition to assumptions (A1) and (A3), their theorem requires that, defining $A_a^2 = E(X^2_{a+m}) + 2 \sum^m_1E(X_{a+m-j}X_{a+m},$ \begin{equation*}\tag{H}\lim_{n\rightarrow\infty} n^{-1}\sum^n_{i = 1} A^2_{a+i} = A^2 \text{exists uniformly in} a (n \rightarrow \infty).\end{equation*} Now it can be shown easily that conditions (A2) and (H) are equivalent in the case of an $m$-dependent sequence satisfying (A1) and (A3). Therefore, a formulation relevant to assumptions (A) is THEOREM 1.1 (Hoeffding-Robbins). If $\{ X_i\}$ is an $m$-dependent sequence satisfying assumptions (A), then it has the central limit property. In the case of a weakly stationary (with mean zero, say) $m$-dependent sequence, the assumptions of the theorem are satisfied except for (A3), which then is a mild additional restriction. For applications in which the existence of moments is not presupposed, e.g., strictly stationary sequences, Theorem 1.1 has been extended by Diananda [6], [7], [8] and Orey [16] in a series of results reducing the moment requirements while retaining the assumption of $m$-dependence. In the present paper the interest is in extensions relaxing the $m$-dependence assumption. A result of Ibragimov [12] in this regard implies THEOREM 1.2 (Ibragimov). If $\{ X_i\}$ is a strictly stationary sequence satisfying assumptions (A) and regularity condition (I), then it has the central limit property. (Condition (I) is defined in Section 2.) Other extensions under condition (I) but not involving stationarity assumptions are Corollary 4.1.3 and Theorem 7.2 below. See also Rosenblatt [17]. Other extensions for strictly stationary sequences, further reducing the dependence restrictions, appear in [12] and [13] and Sections 5 and 6 below. Section 2 is devoted to dependence restrictions. The restrictions (2.1), (2.2) and (2.3), later utilized in Theorem 4.1, are introduced and shown to be closely related to assumptions (A). Although conditional expectations are involved in (2.2) and (2.3), the restrictions are easily interpreted. It is found, under assumptions (A), that if (2.3) is sufficiently stringent, then (2.1) holds in a stringent form (Theorem 2.1). A link between regularity assumptions formulated in terms of joint probability distributions and those involving conditional expectations is established by Theorem 2.2 and corollaries. Implications of condition (I) are given in Theorem 2.3. Section 3 is devoted to the particular dependence restriction (2.1). Theorem 3.1 gives, under assumptions (A), a condition necessary and sufficient for (2.1) to hold in the most stringent form, (3.1). The remaining sections deal largely with central limit theorems. Section 4 obtains the basic result and its general implications. Sections 5, 6 and 7 exhibit particular results for weakly stationary sequences, sequences of martingale differences and bounded sequences. NOTATION AND CONVENTIONS. We shall denote by $\{ X_i\}^\infty_{-\infty}$ a sequence of rv's defined on a probability space. Let $\mathscr{M}_a ^b$ denote the $\sigma$-algebra generated by events of the form $\{(X_{i_1},\cdots, X_{i_k}) \varepsilon E\}$, where $a - 1 < i_1 < \cdots < i_k < b + 1$ and $E$ is a $k$-dimensional Borel set. We shall denote by $\mathscr{P}_a$ the $\sigma$-algebra $\mathscr{M}^a_{-\infty}$ of "past" events, i.e., generated by the rv's $\{ X_a, X_{a-1},\cdots\}$. Conditional expectation given a subfield $\mathscr{B}$ will be represented by $E(\cdot\mid\mathscr{B}),$ which is to be regarded as a function measurable ($\mathscr{B}$). All expectations will be assumed finite whenever expressed.

Let the following expressions denote the binomial and Poisson probabilities, \begin{equation*}\begin{align*}\tag{1.1}B(k; n, p) &= \sum^k_{j=0} b(j; n, p) \\ &= \sum^k_{j=0} \binom{n}{j}p^j (1 - p)^{n-j}, \\ \tag{1.2}P(k; \lambda) &= \sum^k_{j=0}p(k; \lambda) = \sum^k_{j=0} e^{-\lambda}\lambda^k/k\end{align*}!.\end{equation*} Section 2 contains two basic theorems which generalize results of Anderson and Samuels [1] and Jogdeo [7]. These two theorems serve as lemmas for the more detailed results of Sections 3 and 4. Section 3 is devoted to a study of the median number of successes in Poisson trials (i.e. independent trials where the success probability may vary from trial to trial). The study utilizes a method first introduced by Tchebychev [12], generalized by Hoeffding [6], and used by Darroch [5] and Samuels [10]. The results correspond to those for the modal number of successes obtained by Darroch. Ramanujan (see [8]) considered the following equation, where $n$ is a positive integer: \begin{equation*}\tag{1.3}\frac{1}{2} = P(n - 1; n) + y_n p(n; n),\end{equation*} and correctly conjectured that $\frac{1}{3} < y_n < \frac{1}{2}$. In Section 4 we show that for the corresponding binomial equation, \begin{equation*}\tag{1.4}\frac{1}{2} = B(k - 1; n, k/n) + z_{k,n}b(k; n, k/n),\end{equation*} $\frac{1}{3} < z_{k,n} < \frac{2}{3}$ and, for each $k$ and for $n \geqq 2k, z_{k,n}$ decreases to $y_k$ as $n \rightarrow \infty$.

Let $Z_1, Z_2, \cdots, Z_N$ be $N$ random variables with common distribution $P(Z_i \leqq u) = F(u - \theta)$ where $F \varepsilon \mathscr{F}_{cs0}$ which throughout this paper means the class of all continuous distributions symmetric about zero. The distributions considered will furthermore satisfy the regularity conditions of Lemma 3a in [3]. $\theta$ is an unknown constant to be estimated. In the case where $Z_1, \cdots, Z_N$ are independent, the following estimate of $\theta$ has been recently investigated by J. L. Hodges Jr. and E. L. Lehmann [4]. \begin{equation*}\tag{0.1}\theta^{\ast} = \operatorname{med}_{r\leqq s} \{(Z_r + Z_s)/2\},\end{equation*} .e. the median of the $N + \binom{N}{2}$ averages $(Z_r + Z_s)/2$. The asymptotic efficiency of $\theta^{\ast}$ relative to the classical estimate \begin{equation*}\tag{0.2}\hat{\theta} = \sum^N_{i=1} Z_i/N\end{equation*} in the sense of reciprocal ratio of asymptotic variances has been determined in [4] and shown to be $12\sigma^2_z\lbrack\int f{}^2(z) dz\rbrack^2$ where $f$ is the density corresponding to $F$ and $\sigma_z^2$ denotes the variance of the $Z$'s. It follows directly from Theorem 2.2 of [6] that $\theta^{\ast}$, in case of independent but not necessarily symmetrically distributed observations $Z_j$, is a consistent estimate of the pseudomedian of $F$ ([6], p. 178), which in general may be different from the median $\theta$. In this paper we shall consider a situation where only few $(c)$ observations can be collected per day and where the experiments have to be conducted over several $(n)$ days to yield the necessary number of observations. During this period the experimental conditions may easily change, whereby the standard assumption of "independent and identically distributed" observations is violated. The data occur naturally grouped in $n$ blocks, $c$ observations per block, and the possible change of conditions is introduced as a (nuisance) random block effect. We shall study the behavior of the two estimates $\theta^{\ast}$ and $\hat\theta$ under such conditions to find out how robust they are against this kind of dependence. In particular we shall study their asymptotic behavior as $n \rightarrow \infty$ with $c$ fixed, and shall derive a general expression for the asymptotic efficiency of $\theta\ast$ relative to $\hat{\theta}$. The efficiency is finally computed for normal and gross error models.

On the basis of independent samples $\{X_1, \cdots, X_m\}$ and $\{Y_1, \cdots, Y_n\}$ with distributions $F$ and $G$, respectively, the hypothesis that $F \equiv G$ may be tested. Given the functional forms $F(x_1, \cdots, x_m)$ and $G(y_1, \cdots, y_n)$ of the sampling distributions except for values of certain parameters, the likelihood ratio approach, for example, can be used. In this case it is not crucial to assume that the samples are random, i.e., that $F(x_1, \cdots, x_m) = F(x_1) \cdots F(x_m)$ and $G(y_1, \cdots, y_n) = G(y_1) \cdots G(y_n)$, although such a simplification is useful whenever realistic. However, the nonparametric treatment of the problem has relied heavily on the assumption of random samples. Yet if the samples arise as realizations of two stochastic processes, the assumption of randomness is not realistic except in the case of renewal processes. Thus it is desirable to extend the scope of established nonparametric procedures to more general applications. The present paper deals with the Wilcoxon two-sample statistic. Among the desirable features of this statistic, when defined on independent random samples, is its asymptotically normal distribution, which for large samples facilitates a test of the hypothesis that $F \equiv G$ and a calculation of the power for any alternative $(F, G)$. It shall be seen that these aspects are true also when the samples arise from stochastic processes belonging to a wide class, including strictly stationary strongly mixing processes. Assume that the samples $\{X_1, \cdots, X_m\}$ and $\{Y_1, \cdots, Y_n\}$ are independent of each other, but let the random variables within a sample be possibly dependent. Assume that the functions $F(\cdot)$ and $G(\cdot)$ are continuous. The hypothesis $H: F \equiv G$ may be tested (conservatively) by testing the hypothesis $H_0: \gamma = 0$, where $\gamma = 2P\{Y > X\} - 1$. A representation of the Wilcoxon two-sample statistic is the $U$-statistic with sign function as kernel, \begin{equation*}\tag{1.1}U = (mn)^{-1}\sum^m_{i=1} \sum^n_{j=1} s(Y_j - X_i),\end{equation*} where $s(u) = -1, 0, 1$ according as $u < 0, = 0, > 0$. Since $Es(Y - X) = \gamma$, the statistic $U$ affords a natural basis for testing $H_0$. Under appropriate conditions, the statistic $Z = m^{\frac{1}{2}}(U - \gamma)$ has a limiting normal distribution with mean 0 and variance \begin{equation*}\tag{1.2}A^2 = 4 \lim_{k\rightarrow\infty} k^{-1} \operatorname{Var}\lbrack\sum^k_{i=1} G(X_i)\rbrack + 4c \lim_{k\rightarrow\infty}k^{-1} \operatorname{Var}\lbrack\sum^k_{i=1} F(Y_i)\rbrack,\end{equation*} as $m$ and $n \rightarrow \infty$ such that $m/n$ has a limit $c \neq 0$. The main conclusions of this nature are given in Theorems 3.1 and 3.2. Some areas of application are indicated in Section 4. The business of dealing with the quantity $A^2$ is discussed in Section 5. The limiting behavior of $Z$ is obtained by consideration of a statistic asymptotically equivalent in distribution but more amenable to the direct application of central limit theory, an approach put forth by Hoeffding [3] in dealing with a wide class of $U$-statistics as defined on a single sample of mutually independent rv's. The present contribution adapts the method to a single, but important, (two-sample) $U$-statistic with dependence allowed within samples. Define: \begin{equation*}\tag{1.3}W = m^{-\frac{1}{2}} \sum^m_{i=1} \lbrack f_{10}(X_i) - \gamma\rbrack + m^{\frac{1}{2}}n^{-1} \sum^n_{j=1} \lbrack f_{01} (Y_j) - \gamma\rbrack,\end{equation*} where $f_{10}(t) = Es(Y - t) = 1 - 2G(t)$ and $f_{01}(t) = Es(t - X) = 2F(t) - 1$. Since $Ef_{10}(X) = Ef_{01}(Y) = \gamma$, we have $EW = E(Z - W) = 0$. In Section 2 we find conditions such that $E(Z - W)^2 \rightarrow 0$, in which case it follows by Chebyshev's inequality that $(Z - W) \rightarrow 0$ in probability and hence that the statistics $Z$ and $W$ have the same limiting distribution (if any). The application of central limit theory to $W$ is through the sums $\sum^m_1 f_{10}(X_i)$ and $\sum^n_1 f_{01}(Y_j)$, or equivalently through $m^{-\frac{1}{2}} \sum^m_1 G(X_i)$ and $n^{-\frac{1}{2}}\sum^n_1 F(Y_j)$. If each of these independent normed sums has a limiting normal distribution, then $W$ is asymptotically normal, as $m$ and $n \rightarrow \infty$ such that $m/n \rightarrow c \neq 0$. Relevant central limit theorems for sums of dependent variables are utilized in Section 3.

In this paper we consider the discrete time finite state Markov decision problem with Veinott's criterion of maximizing the Cesaro mean of the vector of expected returns received in a finite horizon as the horizon tends to infinity. A necessary and sufficient condition for optimality is obtained, and at the same time we verify Veinott's conjecture that there are optimal stationary policies.

Let $\{X_n, n \geqq 1\}$ be a sequence of independent and identically distributed random variables with mean 0 and finite $\nu$th moment for some $\nu \geqq 2$. Let $S_n = \sum^n_1 X_i$. We observe the $X$'s sequentially and must decide when to stop sampling. If we stop at time $n$ we receive a reward of the form $h_n(S_n)$, and we are concerned with finding stopping rules $t$ which maximize our expected reward, $E\lbrack h_t(S_t)\rbrack$. In particular we are concerned with showing that the so-called "functional equation rule" (FER) (for a definition, see Section 2) is such a (optimal) rule. Chow and Robbins [2] have treated the reward sequence $n^{-1}S_n$ with $X_i = \pm 1$ each with probability $\frac{1}{2}$. Dvoretzky [4] considered reward sequences of the form $n^{-\alpha}S_n (\alpha > \frac{1}{2})$ under the assumptions $EX_i = 0, EX^2_i < \infty$. Teicher and Wolfowitz [7] considered sequences $c_nS_n^\beta (\beta = 1, 2)$ under the same assumptions on the distribution of $X_i$. They impose the conditions $c_n > 0, c^2_{n+1} \leqq c_{n+2}c_n, (n + 1)^\beta c_{n+1} \leqq n^\beta c_n$. We establish certain principles which allow us to relate certain reward sequences $h_n$ of a particularly simple form to others of a more complicated form and in the process conclude that the FER is optimal in the more complicated situation. Using the basic reward sequence $n^{-\alpha}|S_n|^\beta$ with $2\alpha > \beta > 0$ and assuming that $E|X_i|^{\max(2,\beta)} < \infty$, we examine the problem of optimality for reward sequences of the form $c_nS^+_n, c_n|S_n|^\beta, n^{-\alpha} \log|S_n|$, etc., where $c_1, c_2, \cdots$ are constants such that $\lim \sup_{n\rightarrow\infty}n^\alpha c_n < \infty$. It seems somewhat customary in optimal stopping problems to try to verify that the reward sequence (in our case $h_n(S_n))$ is majorized by a non-negative random variable with finite expectation, and then to appeal to one of a class of general theorems in which this regularity condition appears. In the problems we consider we have found it easier to disregard this possibility and to use a direct approach. We begin with a formal presentation of the problem and a development of preliminary results which show that the FER is optimal provided that it is a.s. finite. In later sections we develop and exploit machinery for relating two or more reward sequences and verify that the FER is a.s. finite for the basic reward sequence

Let $X_1, X_2, \cdots$ be independent and identically distributed random variables whose common distribution is of the one-parameter Koopman-Darmois type, i.e., the density function of $X_1$ relative to some $\sigma$-finite nondegenerate measure of $F$ on the real line can be written as $f(x, \theta) = \exp (\theta x - b(\theta))$, where $b(\theta)$ is some real function of the parameter $\theta$. Consider the hypotheses $H_0 = \{\theta \leqq \theta_0\}$ and $H_1 = \{\theta \geqq \theta_1\}$ where $\theta_0 < \theta_1$ and $\theta_0, \theta_1$ are in $\Omega$, the natural parameter space. We want to decide sequentially between the two hypotheses. Suppose $l(\theta)$ is the loss for making a wrong decision when $\theta$ is the true parameter and assume $0 \leqq l(\theta) \leqq 1$ for all $\theta$ and $l(\theta) = 0$ if $\theta$ is in $(\theta_0, \theta_1)$, i.e., $(\theta_0, \theta_1)$ is an indifference zone. Let $c$ be the cost of each observation. It is sufficient to let the decision depend on the sequence $(n, S_n), n \geqq 1$, where $S_n = X_1 + \cdots + X_n$. We shall consider the observed values of $(n, S_n)$ as points in a $(u, v)$ plane. Then, for any test, the region in the $(u, v)$ plane where sampling does not stop is called the continuation region of the test. A test and its continuation region will be denoted by the same symbol. Schwarz [4] introduced an a priori distribution $W$ and studied the asymptotic shape of the Bayes continuation region, say $B_W(c)$, as $c \rightarrow 0$. He showed that $B_W(c)/\ln c^{-1}$ approaches, in a certain sense, a region $B_W$ that depends on $W$ only through its support. Whereas Schwarz's work is concerned with Bayes tests, in this paper the main interest is in characteristics of sequential tests as a function of $\theta$. In particular, it is desired to minimize the expected sample size (uniformly in $\theta$ if possible) subject to certain bounds on the error probabilities. Our approach, like Schwarz's, is asymptotic, as $c \rightarrow 0$. It turns out that an asymptotically optimum test--in the sense indicated above, is $B_W \ln c^{-1}$ if $W$ is a measure that dominates Lebesgue measure. Such a measure will be denoted by $L$ (for Lebesgue dominating) from now on. Thus, Bayes tests, as a tool, will play a significant role in this paper. In order to prove the optimum characteristic of $B_L \ln c^{-1}$, some other results, of interest in their own right, are established. For any $W$ satisfying certain conditions that will be given later, we show that the stopping variable $N(c)$ of $B_W(c)$ approaches $\infty$ a.e. $P_\theta$ for every $\theta$ in $\Omega$. This result together with Schwarz's result that $B_W(c) \ln c^{-1}$ approaches a finite region, leads to the following results: (i) for $B_W(c), E_\theta N(c)/\ln c^{-1}$ tends to a constant for each $\theta$ in $\Omega$ and (ii) the same is true for the stopping variable of $B_W \ln c^{-1}$. Furthermore, it is shown that for $B_L \ln c^{-1}$ the error probabilities tend to zero faster than $c \ln c^{-1}$. Consequently, the contributions of the expected sample sizes of both $B_L \ln c^{-1}$ and $B_L(c)$ to their integrated risks, over any $L$-measure, approach 100%. Moreover $B_L \ln c^{-1}$ is asymptotically Bayes. The last result can be shown without (i) since it is sufficient to show (ii) and to apply the same argument used by Kiefer and Sacks [3] in the proof of their Theorem 1. But we show (i) because of its intrinsic interest and present a different proof using (i). Kiefer and Sacks assumed a more general distribution for $X_1$, constructed a procedure $\delta^{'I}_c$ and showed that it is asymptotically Bayes. Our $B_L \ln c^{-1}$ is somewhat more explicit than their $\delta_c'I$. We would also like to point out that an example of $B_L \ln c^{-1}$, when the distribution is normal, is very briefly discussed in their work. We shall restrict ourselves to a priori distribution $W$ for which $\sup (\mod W)H_0 = \theta_0, \inf (\mod W)H_1 = \theta_1$ and $0 < W(H_0 \cup H_1) < 1$. The phrase "for any $W"$ or "for every $W$" is to be understood in that sense. Any Lebesgue dominating measure satisfies these conditions and also the following type of $W$ that will be used: the support of $W$ consists of $\theta_0, \theta_1$ and a third point $\theta^\ast, \theta_0 < \theta^\ast < \theta_1$. Such a $W$ will be called a $\theta^\ast$-measure, and the corresponding $B_W$ denoted by $B_{\theta^\ast}$. From Schwarz's equations for $B_W$ it follows readily that $B_L \subset B_W$ for every $W$. In particular, $B_L \subset B_{\theta^\ast}$. As a consequence, the statement about the error probabilities as well as others concerning $B_L \ln c^{-1}$ in the last paragraph, remain true when $L$ is replaced by $\theta^\ast$ or any $W$. Those geometric characteristics will be dealt with in Section 2. We shall also show there that $\partial B_{\theta^\ast}$, the boundary of $B_{\theta^\ast}$ (which consists of line segments), is tangent to $\partial B_L$ at some point, and that if $\theta^\ast$ is such that $b'(\theta^\ast) = (b(\theta_1) - b(\theta_0))/(\theta_1 - \theta_0)$ then $\max_{(u, \nu)\text{in}B_L} u = \max_{u,v)\text{in}B_\theta^\ast} u$. Let the ray through the origin and with slope equal to $E_\theta X_1$ intersect $\partial B_L$ at $(m(\theta), m(\theta)E_\theta X_1)$. In Section 3, after proving $\lim_{c\rightarrow 0} N(c) = \infty$ a.e. $P_\theta$, we show $\lim_{c\rightarrow 0} N(c)/\ln c^{-1} = m(\theta)$ a.e. $P_\theta$ and $\lim_{c\rightarrow 0} E_\theta N(c)/\ln c^{-1} = m(\theta)$. It is shown in Section 4 that $\sup_{\theta \text{in} H_0 \mathbf{\cup} H_1} P_\theta$ (error $\mid B_L \ln c^{-1}) = o(c \ln c^{-1})$. The main results are given in Section 5. We first show that after dividing by $c \ln c^{-1}$, the difference of the integrated risks of $B_L \ln c^{-1}$ and $B_W(c)$, for any $W$, tends to zero. It follows from this result that $B_L \ln c^{-1}$ asymptotically minimizes the maximum (over $\theta$ in $\Omega$) expected sample size in $\mathscr{F}(c)$, a family of tests whose error probabilities are bounded by $\max_{i=0,1} P_{\theta_i}$ (error $\mid B_L \ln c^{-1}$). The precise statement is given in Theorem 5.1. A sharper result under a stronger hypothesis is given in Theorem 5.2 which states that $B_L \ln c^{-1}$ asymptotically minimizes the expected sample size $E_\theta N$ for each $\theta, \theta_0 < \theta < \theta_1$, among all procedures of $\mathscr{F}(c)$ for which $E_{\theta_0}N/\ln c^{-1}$ and $E_{\theta_1}N/\ln c^{-1}$ are bounded in $c$.

Let $\{F_n(x)\}$ be a sequence of distribution functions depending on a parameter $n$, and converging to a limiting distribution $\Phi(x)$ as $n$ increases. Then a generalized expansion of Cornish-Fisher type is an asymptotic relation between the quantiles of $F_n$ and $\Phi$. The original Cornish-Fisher formulae [3], [5] provided leading terms of these expansions in the case of normal $\Phi$, expressing a normal deviate in terms of the corresponding quantile of $F_n$ and its cumulants (the "normalizing" expansion) and, conversely, the quantiles of $F_n$ in terms of its cumulants and the corresponding quantiles of $\Phi$ (the "inverse" expansion). The value of both these asymptotic formulae has been well illustrated by their use in approximating the quantiles of complicated distributions (Johnson and Welch [9], Fisher [4], Goldberg and Levine [6]), and for obtaining random quantiles for distribution sampling applications (Teichroew [13], Bol'shev [2]). For a survey of the literature on Cornish-Fisher expansions, and some discussion of their validity, see Wallace ([14], Section 4). In Sections 2, 3 of the present paper, formal expansions are obtained which generalize the Cornish-Fisher relations to arbitrary analytic $\Phi$. Essentially, these expansions provide algorithms for transforming an asymptotic expansion of $F_n$ in terms of the "standard" distribution $\Phi$ into asymptotic relations between the quantiles of these distributions. The "standardizing" expansion of the quantile $u$ of $\Phi$ in terms of the corresponding quantile $x$ of $F_n$ is expressed (Section 2) in terms of a sequence of functions defined by a differential recurrence operator. A similar differential operator appears in the generalized "inverse" expansion for $x$ in terms of $u$ (Section 3), which arises from the application of Lagrange's inversion formula to the equation of quantiles. An asymptotic expansion for quantiles of the Wilks likelihood ratio criterion is given as an example. Formal expansions in terms of the cumulants of $F_n$ and $\Phi$ are obtained in Section 4 by developing $F_n$ about $\Phi$ as a Charlier differential series and collecting terms of like degree in the resulting exponential series. For known cumulants and for normal $\Phi$ these formal expressions reduce, as shown in Section 5, to a general form of the Cornish-Fisher expansions, in which the polynomial terms are represented as sums of products of Hermite polynomials. This representation is shown in Section 6 to account for some properties of the Cornish-Fisher polynomials.

Let $\mathbf{A}_1$ and $\mathbf{A}_2$ be two symmetric matrices of order $p, \mathbf{A}_1$, positive definite and having a Wishart distribution ([2], [23]) with $f_1$ degrees of freedom and $\mathbf{A}_2$, at least positive semi-definite and having a (pseudo) non-central (linear) Wishart distribution ([1], [3], [23], [24]) with $f_2$ degrees of freedom. Now let $\mathbf{A}_2 = \mathbf{CYY}'\mathbf{C}'$ where $\mathbf{Y}$ is $p \times f_2$ and $\mathbf{C}$ is a lower triangular matrix such that $\mathbf{A}_1 + \mathbf{A}_2 = \mathbf{CC}'.$ Now consider the $s$( = minimum $(f_2, p))$ non-zero characteristic roots of the matrix $\mathbf{YY}'$. It can be shown that the density function of the characteristic roots of $\mathbf{Y}'\mathbf{Y}$ for $f_2 \leqq p$ can be obtained from that of the characteristic roots of $\mathbf{YY}'$ for $f_2 \geqq p$ if in the latter case the following changes are made: [23] \begin{equation*}\tag{1.1} (f_1, f_2, p) \rightarrow (f_1 + f_2 - p, p, f_2).\end{equation*} Now, in view of (1.1), we consider only the case $s = p$, based on the density function [12] of $L = \mathbf{YY}'$ for $f_2 \geqq p$. In this paper, some results are obtained first regarding the $i$th elementary symmetric function (esf) of the characteristic roots of a non-singular matrix $\mathbf{P} (\operatorname{tr}_i\mathbf{P})$ which are useful to compute the moments of $\operatorname{tr}_i\mathbf{L}$ and $\operatorname{tr}_i\{(\mathbf{I} - \mathbf{L})^{-1} - \mathbf{I}\}$. In particular, the first two moments of $\operatorname{tr}_2 \mathbf{L}$ are obtained in the non-central linear case. These two moments of the above criteria in the central case have been obtained earlier by Pillai ([18], [19]). Further, from a study of the first four moments of $U^{(p)} = \operatorname{tr}\{(\mathbf{I} - \mathbf{L})^{-1} - \mathbf{I}\}$, [11], [14], two approximations to the distribution of $U^{(p)}$ were obtained in the general non-central case. The approximations are generalizations of those given by Khatri and Pillai [10] for the linear case. The accuracy comparisons of the approximations are also made.

Let $\mathscr{Q} = \{Q(t): 0 \leqq t \leqq 1\}$ be a family of probability distributions on the positive integers parameterized on [0, 1], that is \begin{equation*}\tag{1} P_t(X = x) = q_x(t);\quad x = 1,2,\cdots.\end{equation*} If $G$ is a distribution on $\lbrack 0, 1\rbrack$ the distribution of $X$ is a $G$-mixture over $\mathscr{Q}$ if \begin{equation*}\tag{2} P_G(X = x) = \int^1_0 q_x(t) dG(t) = q_x(G).\end{equation*} $G$ is called the mixing distribution. It is assumed at the outset that the family $\mathscr{Q}$ is known to be identifiable, that is if $q_x(G_1) = q_x(G_2)$ for $x = 1,2,\cdots$, then $G_1 = G_2$. See [12] and [13] for conditions insuring identifiability. Thus it makes sense to attempt to estimate $G$ when one has independent observations on $X$. Some work on estimating $G$ has been done when the mixture is finite [4], [2], [9] and for special $\mathscr{Q}$'s [6], [14]. The problem is of interest not only in an estimation context, but also in the construction of empirical Bayes decision procedures [9]. Our approach is to define a prior distribution on possible values of $G$ and then construct consistent Bayes estimates of $G$ from the posterior distribution. Section 2 gives the needed background on moment spaces, sets up the prior distribution and derives the posterior distribution. In Section 3, the Bayes estimates are defined while Section 4 proves the consistency of the posterior distribution and thus of the estimates. Here, Theorem 1 is not directly applicable to our problem, but is included because of its possible independent interest. Sections 5 and 6 generalize the earlier results.

In many situations, it is of interest to test for the equality of variances or covariance matrices against certain alternatives. Hartley [6] considered the problem of testing for the equality of variances against the alternative that at least one variance is different from the other. Gnanadesikan [3] considered the problem of testing for the equality of variances against the alternative that at least one variance is not equal to the standard. Recently, Krishnaiah [12] considered testing for the equality of variances against the alternative that at least one variance is not equal to the next. In the above procedures, it was assumed that the underlying populations are univariate normal. In this paper, we consider multivariate generalizations of the above test procedures. The procedures proposed in this paper are based upon expressing the total hypothesis as the intersection of some elementary hypotheses and testing these elementary hypotheses by using conditional distributions. In the two sample case, our procedures are similar (but not equivalent) to the procedure proposed by Roy [16]; the test statistics used by him in testing some of the elementary hypotheses are different from those used in this paper.

Let $\mathbf{x}$ and $\mathbf{\theta}$ denote $s$-dimensional column vectors. The components $x_1, x_2,\cdots x_s$ of $\mathbf{x}$ are random variables jointly following an $s$-variate distribution and components $\theta_1, \theta_2,\cdots, \theta_s$ of $\mathbf{\theta}$ are real numbers. The random vector $\mathbf{x}$ is said to follow an $s$-variate Exponential-type distribution with the parameter vector (pv) $\mathbf{\theta}$, if its probability function (pf) is given by \begin{equation*}\tag{1.1} f(\mathbf{x}, \mathbf{\theta}) = h(\mathbf{x}) \exp \{\mathbf{x'\theta} - q(\mathbf{\theta})\},\end{equation*} $\mathbf{x} \varepsilon R_s$ and $\mathbf{\theta} \varepsilon (\mathbf{a}, \mathbf{b}) \subset R_s. R_s$ denotes the $s$-dimensional Euclidean space. The $s$-dimensional open interval $(\mathbf{a}, \mathbf{b})$ may or may not be finite. $h(\mathbf{x})$ is a function of $\mathbf{x}$, independent of $\mathbf{\theta}$, and $q(\mathbf{\theta})$ is a bounded analytic function of $\theta_1, \theta_2,\cdots \theta_s$, independent of $\mathbf{x}$. We note that $f(\mathbf{x}, \mathbf{\theta})$, given by (1.1), defines the class of multivariate exponential-type distributions which includes distributions like multivariate normal, multinomial, multivariate negative binomial, multivariate logarithmic series, etc. This paper presents a theoretical study of the structural properties of the class of multivariate exponential-type distributions. For example, different distributions connected with a multivariate exponential-type distribution are derived. Statistical independence of the components $x_1, x_2,\cdots, x_s$ is discussed. The problem of characterization of different distributions in the class is studied under suitable restrictions on the cumulants. A canonical representation of the characteristic function of an infinitely divisible (id), purely discrete random vector, whose moments of second order are all finite, is also obtained. $\varphi(\mathbf{t}), m(\mathbf{t}), k(\mathbf{t})$ denote, throughout this paper, the characteristic function (ch. f.), the moment generating function (mgf), and the cumulant generating function (cgf), respectively, of a random vector $\mathbf{x}$. The components $t_i$ of the $s$-dimensional column vector $\mathbf{t}$ are all real.

A new method, simpler than previous methods due to Chung (1954) and Sacks (1958), is used to prove Theorem 2.2 below, which implies in a simple way all known results on asymptotic normality in various cases of stochastic approximation. Two examples of application are concerned with Venter's (1967) extension of the RM method and Fabian's (1967) modification of the KW process. Previously, although there was no difficulty in adopting one or the other method, the proofs in various cases had to be done almost ab initio or skipped leaving a gap (see Venter (1967)). The new proof is similar to that of Chung except that the basic recurrence relation is used to obtain the asymptotic characteristic function rather than limits of all moments. We remark that Lemma 2.1, a simple corollary to Chung's lemma is used only to obtain condition (2.2.4) which is weaker than (2.2.3) if $\alpha = 1$ and which corresponds to the usual Lindeberg condition. Both conditions (2.2.3) and (2.2.4) are weaker than the corresponding condition (3.4) in Sacks (1958). In what follows $(\Omega, \mathscr{S}, P)$ will be a probability space, relations between and convergence of random variables, vectors, and matrices will be meant with probability one unless specified otherwise. We shall write $X_n \sim \mathscr{L}$ if $X_n$ is asymptotically $\mathscr{L}$-distributed and $X_n \sim Y_n$, for two sequences of random vectors, if for any $\mathscr{L}, X_n \sim \mathscr{L}$ if and only if $Y_n \sim \mathscr{L}$. The indicator function of a set $A$ will be denoted by $\chi A$, the expectation and conditional expectation by $E$ and $E_F$, respectively. $R^k$ is the $k$-dimensional Euclidean space the elements of which are considered to be column vectors, $R = R^1, R^{k\times k}$ is the space of all real $k \times k$ matrices. The symbols $\mathbf{R}, \mathbf{R}^k, \mathbf{R}^{k\times k}$, denote sets of all measurable transformations from $(\Omega, \mathscr{S})$ to $R, R^k R^{k\times k}$, respectively. The unit matrix in $R^{k\times k}$ is denoted by $I$ and $| |$ is the Euclidean norm. With $h_n$ a sequence of numbers, $o(h_n), O(h_n), o_u(h_n), O_u(h_n)$ denote sequences $g_n, G_n, q_n, Q_n$, say, of elements in one of the sets $\mathbf{R}, \mathbf{R}^k, \mathbf{R}^{k\times k}$ such that $h_n^{-1} g_n \rightarrow 0, | h_n^{-1} G_n|\leqq f$ for an $f \varepsilon \mathbf{R}$ and all $n, h_n^{-1} q_n \rightarrow O$ uniformly on a set of probability one, $| h_n^{-1} Q_n| \leqq K$ for a $K \varepsilon R$ and all $n$. In special cases $o(h_n)$ may be constant on $\Omega$ and considered as a sequence with elements in $R, R^k$ or $R^{k\times k}$. Similarly in other cases. For Chung's lemma, which will be frequently referred to, or used later without reference, see Fabian ((1967), Lemma 4.2); note that it holds with $\beta = 0$, too.

Let $X = (X_t, \mathbf{F}_t, P^x)_{t\geqq0}$ be a Markov process where $(X_t; t \geqq 0)$ is the trajectory or sample path, $\mathbf{F}_t$ is the definitive $\sigma$-algebra of events generated by $(X_s; 0 \leqq s \leqq t)$, and $P^x$ is the probability distribution on sample paths corresponding to an initial state $x$. The state space is taken as the semi-compact $(E, \mathbf{C})$ where $E$ is a locally compact separable metric space with family of open sets $\mathbf{C}$. A non-negative extended real valued random variable $T$ such that for each $t \geqq 0, \{T \leqq t\} \varepsilon \mathbf{F}_t$ is called a Markov time or stopping time. This paper studies the problem of choosing a stopping time $T$ which, for a fixed $\lambda \geqq 0$, maximizes one of the following criteria:\begin{equation*}\tag{1} \Theta_T(x) = E^xe^{-\lambda T} g(X_T);\end{equation*}\begin{equation*}\tag{2} \lambda_T(x) = E^x\lbrack e^{-\lambda T} g(X_T) - \int^T_o e^{-\lambda s} c(X_s) ds\rbrack, \text{where} E^xT < \infty;\text{or}\end{equation*}\begin{equation*}\tag{3} \Phi_T(x) = E^x\lbrack g(X_T) - \int^T_0 c(X_s) ds\rbrack/E^xT, \text{where} 0 < E^xT < \infty;\end{equation*} where $g$ and $c$ are non-negative continuous functions defined on the state space of the process. Dynkin [9] studied criterion (1) where $\lambda = 0$ under the general assumption that $X$ is a standard process with a possibly random lifetime and under very weak continuity assumptions concerning the return function $g$. He showed that criterion (2) can often be transformed into criterion (1), and thus his approach is applicable in this case as well. This paper studies optimal stopping in a Markov process having a Feller transition function, a special case in Dynkin's development. We further specialize to exponentially distributed lifetimes which causes the appearance of a discount factor $e^{-\lambda t}$, with the natural interpretation that a dollar transaction $t$ time units hence has a present value of $e^{-\lambda t}$. Criterion (3) often has the meaning of a long-run time average return and a means of transforming this criterion into criterion (2) is given. Finally, some techniques for implementing Dynkin's approach in a variety of commonly occurring situations are given along with examples of their use.