We give simple necessary and sufficient conditions on the mean and covariance for a Gaussian measure to be equivalent to Wiener measure. This was formerly an unsolved problem [26]. An unsolved problem is to obtain the Radom-Nikodym derivative $d\mu/d\nu$ where $\mu$ and $\nu$ are equivalent Gaussian measure [28]. We solve this problem for many cases of $\mu$ and $\nu$, by writing $d\mu/d\nu$ in terms of Fredholm determinants and resolvents. The problem is thereby reduced to the calculation of these classical quantities, and explicit formulas can often be given. Our method uses Wiener measure $\mu_w$ as a catalyst; that is, we compute derivatives with respect to $\mu_w$ and then use the chain rule: $d\mu/d\nu = (d\mu/d\mu_w)/(d\nu/d\mu_w)$. Wiener measure is singled out because it has a simple distinctive property--the Wiener process has a random Fourier-type expansion in the integrals of any complete orangeade system. We show that any process equivalent to the Wiener process $W$ can be realized by a linear transformation of $W$. This transformation necessarily involves stochastic integration and generalizes earlier nodulation transformations studied by Legal [21] and others [4], [27]. New variants of the Wiener process are introduced, both conditioned Wiener processes and free $n$-fold integrated Wiener processes. We given necessary and sufficient conditions for a Gaussian process to be equivalent to any one of the variants and also give the corresponding Radon-Niels (R-N) derivative. Last, some novel uses of R-N derivatives are given. We calculate explicitly: (i) the probability that $W$ cross a slanted line in a finite time, (ii) the first passage probability for the process $W(T + 1) - W(t)$, and (iii) a class of function space integrals. Using (iii) we prove a zero-one law for convergence of certain integrals on Wiener paths.

This paper redefines the concept of sampling from a population with a given parametric form, and thus leads up to some proposed alternatives to the existing Bayesian and fiducial arguments for deriving posterior distributions. Section 2 spells out the basic assumptions of the suggested class of sampling models, and Section 3 suggests a mode of inference appropriate to the sampling models adopted. A novel property of these inferences is that they generally assign upper and lower probabilities to events concerning unknowns rather than precise probabilities as given by Bayesian or fiducial arguments. Sections 4 and 5 present details of the new arguments for binomial sampling with a continuous parameter $p$ and for general multinominal sampling with a finite number of contemplated hypotheses. Among the concluding remarks, it is pointed out that the methods of Section 5 include as limiting cases situations with discrete or continuous observable and continuously ranging parameters.

Let $0 < r < 1$. Let $\mu$ be a probability on $S$, the closed unit square, which assigns probability 1 to the vertical line with abscissa $r$. Let $\Delta$ be the set of distribution functions on the closed unit interval $I$. Endow $\Delta$ with its weak $\ast$ Borel $\sigma$-field. For any closed sub-interval $H$ of $I$, let $\langle H\rangle$ be the linear function which maps $I$ onto $H$ and sends 0 to the left endpoint of $H$; and $H_0$ (respectively, $H_1$) be the image of $\lbrack 0, r\rbrack$ (respectively, $\lbrack r, 1\rbrack$) under $\langle H\rangle$. Write $B$ for the set of all finite sequences of 0's and 1's (including the empty sequence $\varnothing$). For $b \varepsilon B$ and $\epsilon = 0$ or 1, $I_{b\epsilon} = (I_b)_\epsilon$, where $I_\varnothing = I$. For $G \varepsilon \Delta$, let $|G|$ be the unique probability on $I$ with $G(x) = |G|\lbrack 0, x\rbrack$ for all $x \varepsilon I$. Let $Y_b :b \varepsilon B$ be independent random variables, such that $(r, Y_b)$ has distribution $\mu$. Let $P_\mu$, a probability on $\Delta$, be the distribution of the (random) $F \varepsilon \Delta$ satisfying: $|F|(I_\varnothing) = 1$ and $|F|(I_{b0}) = Y_b |F|(I_b)$ for all $b \varepsilon B$. For a more detailed description, see [2], or Sections 1 and 2 of [3]. Say $F \varepsilon \Delta$ is strictly singular with respect to $G \varepsilon \Delta$ if there is no $x$ for which the ratio of $F(x + h) - F(x)$ to $G(x + h) - G(x)$ converges to a finite, positive limit as $h$ tends to 0. The object of this note is to prove the Theorem. Let $0 < r < 1$. Let $\mu$ and $\nu$ be distinct probabilities on $S$, both assigning measure 1 to the vertical line with abscissa $r$. Then there are weak $\ast$ Borel subsets $C$ and $D$ of $\Delta$, with $P_\mu(C) = P_\nu(D) = 1$, and each distribution function in $C$ strictly singular with respect to each distribution function in $D$. Let $\mu^\ast = \mu\{(r,0)\} + \mu\{(r,1)\}$, and $\nu^\ast = \nu\{(r,0)\} + \nu\{(r,1)\}$. The easy case (either $\mu^\ast = 1$ or $\nu^\ast = 1$) of the Theorem is proved in Section 5. The harder case ($\mu^\ast < 1$ and $\nu^\ast < 1$) is proved in Section 4. Section 3 contains preliminary material on branching processes, which may be of general interest.

The current investigation is a natural outgrowth of [2], being concerned with the variance of stopping rules and the effect of non-zero means on the variance of a randomly stopped sum. Some martingale generalizations of applications of [2] also appear.

Associated with certain of the learning models introduced by Bush and Mosteller [1] are random walks $p_1, p_2, p_3, \cdots$ on the closed unit interval with transition probabilities of the form \begin{equation*}\tag{1}P\lbrack p_{n + 1} = p_n + \theta_1(1 - p_n) \mid p_n\rbrack = \varphi(p_n)\end{equation*} and \begin{equation*}\tag{2}P\lbrack p_{n + 1} = p_n - \theta_2p_n \mid p_n\rbrack = 1 - \varphi(p_n)\end{equation*} where $0 < \theta_1, \theta_2 < 1$ and $\varphi$ is a mapping of the closed unit interval into itself. In the experiments to which these models are applied, response alternatives $A_1$ and $A_2$ are available to a subject on each of a sequence of trials, and $p_n$ is the probability that the subject will make response $A_1$ on trial $n$. Depending on which response is actually made, one of two events $E_1$ or $E_2$ ensues. These events are associated, respectively, with the increment $p_n \rightarrow p_n + \theta_1(1 - p_n)$ and the decrement $p_n \rightarrow p_n - \theta_2p_n$ in $A_1$ response probability. The conditional probabilities $\pi_{ij}$ of event $E_j$ given response $A_i$ do not depend on the trial number $n$. Thus (1) and (2) are obtained with $\varphi(p) = \pi_{11}p + \pi_{21}(1 - p)$. Since the linearity of the functions $\varphi$ which arise in this way is of no consequence for the work presented in this paper, we will assume instead simply that \begin{equation*}\tag{3}\varphi \varepsilon C^2(\lbrack 0, 1\rbrack).\end{equation*} We impose one further restriction on $\varphi$ which excludes some cases of interest in learning theory: \begin{equation*}\tag{4}\epsilon_1 = \min_{0 \leqq p \leqq 1} \varphi(p) > 0 \text{and} \epsilon_2 = \max_{0 \leqq p \leqq 1} \varphi(p) < 1.\end{equation*} It follows from a theorem of Karlin ([5], Theorem 37) that under (1)-(4) the distribution function $F^{(n)}_{\theta_1,\theta_2,\varphi}$ of $p_n$ (which depends, of course, on the distribution $F$ of $p_1$) converges as $n$ approaches infinity to a distribution $F_{\theta_1,\theta_2,\varphi}$ which does not depend on $F$. It is with the distributions $F_{\theta_1,\theta_2,\varphi}$ that the present paper is concerned. Very little is known about distributions of this family, though some results may be found in Karlin [5], Bush and Mosteller [1], Kemeny and Snell [6], Estes and Suppes [3], and McGregor and Hui [8]. The only theorem in the literature directly relevant to the present work is one of McGregor and Zidek [9] as a consequence of which, in the case $\theta_1 = \theta_2 = \theta, \varphi(p) \equiv \frac{1}{2}$, $\lim_{\theta \rightarrow 0} \lim_{n \rightarrow \infty} P\lbrack\theta^{-\frac{1}{2}}(p_n - \frac{1}{2}) \leqq x\rbrack = \Phi(8^{\frac{1}{2}}x)$ where $\Phi$ denotes the standard normal distribution function; that is, the distribution $F_{\theta,\theta,\frac{1}{2}}(\theta^{\frac{1}{2}}x + \frac{1}{2})$ converges to a normal distribution as the "learning rate" parameter $\theta$ tends to 0. We will prove, by means of another method, that this phenomenon is of much greater generality. Theorem 1 below shows that, for any positive constant $\zeta$ and any $\varphi$ with $\max_{0 \leqq p \leqq 1}\varphi'(p) < \min (1, \zeta)/\max (1, \zeta)$ there is a constant $\rho$ such that $F_{\theta,\zeta\theta,\varphi}(\theta^{\frac{1}{2}}x + \rho)$ converges to a normal distribution as $\theta \rightarrow 0$. A nonnormal limit is obtained if $\theta_1$ approaches 0 while $\theta_2$ remains fixed as is shown in Theorem 2. In this case $F_{\theta_1,\theta_2,\varphi}(\theta_1x)$ converges to an infinite convolution of geometric distributions. If $f(p,\theta) = p + \theta(1 - p)$ then (1) and (2) can be written in the form \begin{equation*}\tag{5}P\lbrack p_{n + 1} = f(p_n, \theta_1) \mid p_n\rbrack = \varphi(p_n)\end{equation*} and \begin{equation*}\tag{6}P\lbrack p_{n + 1} = 1 - f(1 - p_n, \theta_2) \mid p_n\rbrack = 1 - \varphi(p_n).\end{equation*} In Section 4 it is shown that the linearity of $f(p,\theta)$ in $p$ and $\theta$ is not essential to the phenomena discussed above. Theorems 3 and 4 present generalizations of Theorems 2 and 1, respectively, to "learning functions" $f(p,\theta)$ subject only to certain fairly weak axioms. A somewhat different learning model, Estes [2] $N$-element pattern model, leads to a finite Markov chain $p_1, p_2, p_3, \cdots$ with state space $S_N = \{jN^{-1}: j = 0, 1, \cdots, N\}$ and transition probabilities \begin{align*}\tag{7}P\lbrack p_{n + 1} &= p_n + N^{-1} \mid p_n\rbrack = \varphi(p_n), \\ \tag{8}P\lbrack p_{n + 1} &= p_n - N^{-1} \mid p_n\rbrack = \psi(p_n),\\ \end{align*} and \begin{equation*}\tag{9}P\lbrack p_{n + 1} = p_n \mid p_n\rbrack = 1 - \varphi(p_n) - \psi(p_n)\end{equation*} where $\varphi(p) = c\pi_{21}(1 - p), \psi(p) = c\pi_{12}p, 0 < c \leqq 1$, and for the sake of this discussion we suppose that $0 < \pi_{12}, \pi_{21} < 1$. In this case a limiting distribution $F_{N, \varphi,\psi}$ of $p_n$ as $n \rightarrow \infty$ exists and is independent of the distribution of $p_1$ by a standard theorem on Markov chains. Estes [2] showed that the limit is binomial over $S_N$ with mean $r = \pi_{21}/(\pi_{12} + \pi_{21})$. It then follows from the central limit theorem that $\lim_{N \rightarrow \infty}\lim_{n \rightarrow \infty} P\lbrack N^{\frac{1}{2}}(p_n - r) \leqq x\rbrack = \Phi\lbrack x/(r(1 - r))^{\frac{1}{2}}\rbrack.$ In Section 5 it is shown that our method permits an extension of this result to much more general $\varphi$ and $\psi$.

If $P$ is a transient Markov chain having the invariant measure $\mu$, and if at time 0 particles are distributed in the state space $\Omega$ according to the Poisson law, with mean $\mu(x)$ at $x$, and these particles are then allowed to move independently of the others according to the law $P$, the system maintains itself in macroscopic equilibrium. In this paper we investigate several phenomena connected with this system.

Simultaneous consideration of $N$ statistical decision problems having identical generic structure constitutes a compound decision problem. The risk of a compound decision problem is defined as the average risk of the component problems. When the component decisions are between two fully specified distributions $P_1$ and $P_2, P_1 \neq P_2$, Hannan and Robbins [5] give a decision function whose risk is uniformly close (for $N$ large) to the risk of a best "simple" procedure based on knowing the proportion of component problems in which $P_2$ is the governing distribution. This result was motivated by heuristic arguments and an example (component decisions between $\mathfrak{N}(-1, 1)$ and $\mathfrak{N}(1, 1)$ given by Robbins [8]. In both papers, the decision functions for the component problems depended on data from all $N$ problems. This paper generalizes and strengthens a result of Hannan and Robbins (Theorem 4, [5]) to the case where each component problem consists of making one of $n$ decisions based on an observation from one of $m$ distributions. Specifically, we find upper bounds for the difference in the risks (the regret function) of a certain compound procedure and a best "simple" procedure which is Bayes against the empirical distribution on the component parameter space. Theorem 2 gives sufficient conditions for a uniform (in parameter sequences) bound on the regret function of order $N^{-\frac{1}{2}}$, while Theorem 3 states sufficient conditions for a uniform bound of order $N^{-1}$. For $m = n = 2$, Theorem 2 furnishes a strengthening of Theorem 4 of [5]. More extensive results for the case $m = n = 2$ are given in a paper by Hannan and Van Ryzin [6]. Please note that the case considered here makes the $N$-decisions after the data from all $N$ problems are available. The sequential case ($k$th decision after observations $1, 2, \cdots, k, k = 1, \cdots, N$) is treated by Hannan in [3] and by Samuel in [10].

$T$ is the sequential probability ratio test (SPRT) based on the sequence $\{X_n\}$ whose family of distributions $\{P_\theta, \theta \varepsilon \Theta\}$ satisfies certain sufficiency, monotone likelihood ratio, and consistency assumptions. Sufficiency reduces the criterion of $T$ to $q\theta_2^n/q\theta_1^n$, where $q_{\theta n}$ is the density of $X_n, \theta_1$ and $\theta_2$ are the values of $\theta$ specified by the hypothesis and alternative respectively, and $\theta_1 < \theta_2$. It is assumed that $q_{\theta n}(x)$ has the asymptotic form: $q_{\theta n}(x) \sim f_{\theta n}(x) \equiv K(n)C(\theta, x)e^{nh(\theta, x)}$ as $n \rightarrow \infty$. The Simplified SPRT $T^\ast$ is proposed where $T^\ast$ uses the criterion $e^{nh(\theta_2,\cdot)}/e^{nh(\theta_1,\cdot)}$. The following conditions are relevant: Condition C states that $h(\theta,x)$ has a unique maximum at $x = \theta$ and $h(\theta,\theta)$ is free of $\theta$; Condition D(i) requires $\Delta_{\theta n}(x) \equiv q_{\theta n}(x)/f_{\theta n}(x)$ to be bounded for all $\theta, x$, and $n$; and Condition D(ii) states that for each $\theta,\Delta_{\theta n}(x) \rightarrow 1$ uniformly in $x$ for $x$ in a neighborhood of $x = \theta$. Let $N$ and $N^\ast$ be the sample sizes of $T$ and $T^\ast$ respectively, and let $\theta_0$ be the solution of $h(\theta_2, x) = h(\theta_1, x)$. It is shown in Section 3 that Conditions C and D imply: $P_\theta(N^\ast > n) < \gamma\delta^n/n^{\frac{1}{2}}$, where $0 < \delta < 1, \gamma < \infty$ and $\theta \neq \theta_0$. The same is true for $N$. Thus, the moment generating functions of $N$ and $N^\ast$ exist, and inequalities for the expected values of $N$ and $N^\ast$ are readily obtained with respect to any $P_\theta, \theta \neq \theta_0$. The following monotonicity properties of $E_\theta N$ and $E_\theta N^\ast$ are established under an additional condition in Section 4: the expected values increase for $\theta$ less than a certain interval containing $\theta_0$ and decrease for $\theta$ greater than this interval. Several examples are discussed in Section 5, and the conditions are checked in Section 6.

The purpose of this paper is to show that a certain class of experiments is minimal essentially complete asymptotically and to demonstrate that this is not generally the case for finite sample sizes. The model to be discussed involves linear experiments consisting of uncorrelated observations. That is, the experimenter may choose $n$ uncorrelated random variables $Y(x_1), Y(x_2), \cdots, Y(x_n)$ with expectation, \begin{equation*}\tag{1.1}E\lbrack Y(x)\rbrack = x'\theta\end{equation*} where $\theta$ is a parameter vector in $k$ dimensional space $E^{(k)}$ and $x$ is a vector also in $E^{(k)}$, to some extent to be chosen by the experimenter, and the variance $V\lbrack Y(x)\rbrack = \sigma^2$. An experiment, $e$, of size $n$ is fully specified, using the terminology of Elfving [6], by a spectrum $(x_1, x_2, \cdots, x_r)$ consisting of the different $x$'s and an allocation $(n_1, n_2, \cdots, n_r)$ where $n_1 + n_2 + \cdots + n_r = n$. Thus, $e$ can be represented by $e = (n_1, n_2, \cdots, n_r; x_1, x_2, \cdots, x_r)$. We are concerned with the case where $x$ is restricted to lie in a set $A$. The questions to be discussed relate to the choice of the $x$'s in $A$. Some results in this direction have previously been obtained and are discussed by Elfving [5], [6], [7], Ehrenfeld [2], [3] and Kiefer [9], [10]. The information matrix, $F(e)$, associated with experiment $e$, is given by \begin{equation*}\tag{1.2}F(e) = n_1x_1x_1' + \cdots + n_rx_rx_r' = n(p_1x_1x_r' + \cdots + p_rx_rx_r')\end{equation*} where $p_j = n_j/n$ is called the relative allocation at $x_j$. Also, $p_j \geqq 0$ and $p_1 + p_2 + \cdots + p_r = 1$. Strictly speaking, in the exact theory, the $p_j$'s can only range over multiples of $1/n$. However, in the approximate or asymptotic theory, the $p_j$'s have been allowed to range continuously from 0 to 1, see Elfving [5], [6] and Kiefer [9], [10]. We will show later that the exact and asymptotic theory differ in essential ways. We will assume throughout that the set $A$ is symmetric about the origin. This can be done, without essential restriction, since $-Y(x)$ is automatically available with $Y(x)$, see Elfving [6]. Before stating some results we have to introduce some notation for certain classes of experiments. In the approximate theory we denote by $\mathfrak{E}\lbrack A\rbrack$ the set of experiments $e$ with the $x$'s restricted to be in $A$. We assume that any $e \varepsilon \mathfrak{E}\lbrack A\rbrack$ is described by a spectrum $(x_1, x_2, \cdots, x_r)$ consisting of a finite number of $x$'s in $A$ and a relative allocation $(p_1, p_2, \cdots, p_r)$. The sample size plays no role here since we are dealing with the asymptotic case. Also, the value of $r$ is part of the choice of the experiment. In the exact theory, we denote by $\mathfrak{E}_N\lbrack A\rbrack$ the set of experiments $e$ where the $x$'s are restricted to be in set $A$ and the sample size $n \leqq N$. To compare experiments, we introduce a partial ordering $e_1 \geqq e_2$ which will mean that \begin{equation*}\tag{1.3}V_{e_1}\lbrack t'\hat\theta\rbrack \leqq V_{e_2}\lbrack t'\hat\theta\rbrack\quad \text{for all} t\end{equation*} where $V_e\lbrack t'\hat\theta\rbrack$ denotes the variance of the least square estimate of $t'\theta$ with experiment $e$. When $t'\theta$ is not estimable with respect to $e, V_e\lbrack t'\hat\theta\rbrack$ is set equal to infinity. It was shown, Ehrenfeld [3], that relation (1.3) is equivalent to $F(e_1) - F(e_2)$ being a non-negative definite matrix. Finally, we consider comparing two classes of experiments $\mathfrak{E}_1$ and $\mathfrak{E}_2$. We say that $\mathfrak{E}_1$ is essentially complete with respect to $\mathfrak{E}_2$ when for any $e_2 \varepsilon \mathfrak{E}_2$ there exists an $e_1 \varepsilon \mathfrak{E}_1$ such that $e_1 \geqq e_2$. Furthermore, $\mathfrak{E}_1$ is said to be a minimal essentially complete class when no proper subset of $\mathfrak{E}_1$ is complete with respect to $\mathfrak{E}_2$. We also define a weaker kind of minimality, designated by minimal essential completeness ($W$), in the exact or asymptotic case. Suppose $\mathfrak{E}$, depends on a set $B$ of experimental points. The essentially complete class $\mathfrak{E}_1$, is minimal essentially complete $(W)$ with respect to $\mathfrak{E}_2$. That is, one cannot remove any elements of $B$ without losing essential completeness. Various results concerning essentially complete classes were proven and discussed by Ehrenfeld [3], Elfving [6] and Kiefer [9], [10]. Let $E(A)$ denote the extreme points of the set $A$. A point $x$ is an extreme point of $A$ if it is not a convex combination of two or more points of the set $A$. We will show when $A$ is compact that, in the asymptotic case, $\mathfrak{E}\lbrack E(A)\rbrack$ is minimal essentially complete $(W)$ with respect to $\varepsilon \lbrack A\rbrack$. Furthermore, we will demonstrate, with an example and some general theorems, that in the exact case $\mathfrak{E}_N\lbrack E(A)\rbrack$ is not necessarily even essentially complete with respect to $\mathfrak{E}_N\lbrack A\rbrack$.

In a previous paper [2], the risk (essentially, the variance) of certain mean estimators that in some way allow for a possible occurrence of stragglers (random variables with different mean or larger variance) have been numerically computed when the respective variances of stragglers and nonstragglers are given. This restriction is relaxed in the present paper; only the ratio of these variances is assumed to be known. A variety of cases as explained in detail in Sections 2 and 3 is being considered: The underlying distribution may be a normal (Gaussian), a logistic, or a generalized Cauchy distribution; there may be none, one, or two stragglers; the ratio of the standard deviations may be 3 or 6; finally, trimmed and Winsorized means and Bayes estimators are considered. The results are discussed in Section 5. They support the suggestion (J. W. Tukey [3]) to trim the sample by all observations that deviate substantially from the sample mean and to Winsorize those observations that deviate moderately, but trimming off exactly two observations is almost without reservations a strategy superior to always Winsorizing two. A very satisfactory behavior is exhibited by some estimators that formally are Bayes solutions for the Gaussian distribution but that also have been used with the other ones. This suggests to further pursue estimators with similar properties.

This paper is concerned with consistent estimates of a quantile of a distribution function when the quantile is not unique. To be more precise, since the quantile is assumed not to be unique, we are concerned with obtaining a consistent estimate of the smallest $p$th quantile for a fixed $p(0 < p < 1)$, and from this procedure we can estimate the largest $p$th quantile. In Section 2 we consider the oscillating character and limit distribution of the sample $p$th quantile. Also included in this section is a precise statement of the problem to be solved. In Section three the problem of medians only is considered. Here we treat the sample median of the set of averages of all $\binom{n + 1}{2}$ pairs of observations $X_1, \cdots, X_n$, which is briefly mentioned in [1]. We give a proof that this sample median converges almost surely to the center median of the original population, provided that the original distribution function is symmetric about a median. If this symmetry condition is relaxed, it is shown that this sample median of averages of pairs need not converge; and even if it did converge, it might converge to a number which is not a median of the parent distribution. In Section 4, strongly consistent estimates of the smallest $p$th quantile are obtained (for fixed $p, 0 < p < 1$) which do not depend on the functional form of the parent distribution function, and a characterization of weakly consistent estimates is given.

Let $y$ be a single observation on a $p \times 1$ random vector which is distributed according to the multivariate normal distribution with mean vector $\theta$ and covariance matrix $I$. Consider the problem of estimating $\theta$ when the loss function is the sum of the squared errors in estimating the individual components of $\theta$. Let $G$ be a $p \times p$ real matrix. Then we will prove that the estimate $Gy$ is admissible if and only if $G$ is symmetric, and the characteristic roots of $G, g_i$ say, $i = 1, 2, \cdots, p$, satisfy $0 \leqq g_i \leqq 1$, with equality at one for at most two of the roots. The proof concerning the characteristic roots uses the results of Karlin [4] and James and Stein [3]. We give two proofs of the fact that $Gy$ is inadmissible when $G$ is asymmetrical. In the first proof we give an estimate $G^\ast y$ that is better than $Gy$, where $G^\ast$ is a symmetric matrix. This not only adds to the practicality of the result, but also enables us to resolve the question of which estimates are admissible in the restricted class of estimates of the form $Gy$. The method of the second proof, which utilizes a theorem of Sacks [6], leads to the following finding: If $Gy$ is admissible, then $Gy$ must be a generalized Bayes procedure, where the unique generalized prior distribution must be either a multivariate normal distribution with mean vector zero and a specified covariance matrix determined by $G$, or the product of a distribution which is multivariate normal over a subspace of the parameter space and a distribution which is uniformly distributed over a subspace of the parameter space. This latter finding generalizes the well known one dimensional case. In Section 2 then the main results are proved. In Section 3 some remarks concerned with generalizing the main results are given. We remark here that the decision theory terminology used is more or less that of Blackwell and Girshick [1].

Let $x$ be a random $p \times 1$ column vector having a multivariate normal distribution with unknown mean vector $\mu$ and unknown covariance matrix $\Sigma$. We wish to test the hypothesis of "sphericity," namely $H:\Sigma = \sigma^2I_p$, where $\sigma^2 > 0$ is an unknown positive constant. Alternatives to $H$ which are considered are $H_A : \Sigma$ positive definite, but $\Sigma \neq \sigma^2I$. Given $N$ observation vectors $x^{(1)}, x^{(2)}, \cdots, x^{(N)}$, independently distributed, each with the distribution of $x$, we can reduce consideration to the sufficient statistic $(\bar x, S)$, where $\bar x = N^{-1} \sum^N_{i = 1} x^{(i)},\quad S = \sum^N_{i = 1} (x^{(i)} - \bar x)(x^{(i)} - \bar x)'.$ Then $\bar x$ has a multivariate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma/N$, and $S$ has the Wishart distribution, i.e., has density \begin{equation*}\tag{1.1}p(S) = C_{p,n} |S|^{(n - p - 1)/2} |\Sigma|^{-n/2} \exp \lbrack -\frac{1}{2} \mathrm{tr} \Sigma^{-1}S\rbrack,\quad S > 0\end{equation*} where $C^{-1}_{p,n} = \pi^{p(p - 1)/4}2^{np/2} \prod^p_{i = 1} \Gamma((n - i + 1)/2),\quad p \leqq n,$ and $n = N - 1$. Henceforth we shall denote the fact that a random matrix $Z$ has the density (1.1) by writing $\mathfrak{L}(Z) = \mathfrak{W}(\Sigma, p, n)$; thus, $\mathfrak{L}(S) = \mathfrak{W}(\Sigma, p, n)$. Mauchly [4] has found the likelihood ratio test for $H$ v.s. $H_A$. The rejection region of this test can be written in the form: \begin{equation*}\tag{1.2}T(S) \equiv (\mathrm{tr} S)^p/|S| > K,\end{equation*} where $T(S)/p^p$ is the $-2/N$th power of the likelihood ratio statistic $\lambda$. The moments of the likelihood ratio statistic $\lambda$ under $H$ were obtained by Mauchly [4]. Anderson [1] uses these moments to give the exact distribution of $\lambda$ under $H$ and to obtain an asymptotic expansion of this null distribution. The distribution of $\lambda$ under $H_A$ has been obtained for the case $p = 2$ by Girshick [3], but the distribution of $\lambda$ under $H_A$ for $p > 2$ appears to be highly untractable. In this note, we show that the distribution of $T(S)$ is related to the distribution of Bartlett's statistic for testing homogeneity of variances (viz., Anderson [1]). From this relation, we derive that Mauchly's test (1.2) is unbiased. A derivation of the asymptotic distribution of $T(S)$ under $H_A$ completes the note. It should be mentioned here that a direct relationship between the likelihood ratio statistic $\lambda$ and the Bartlett statistic for testing the homogeneity of variances for the elements of $x$ is given by Anderson [1]. He shows that $H:\Sigma = \sigma^2I$ is a combination of two hypotheses $H_1:\Sigma$ is diagonal, and $H_2:\Sigma = \sigma^2I$ given that $\Sigma$ is diagonal. Hypothesis $H_2$ is the hypothesis of the homogeneity of the variances of the elements of the vector $x$ given that these random elements are stochastically independent. The likelihood ratio statistic $\lambda_2$ for testing this hypothesis is a monotone function of Bartlett's statistic. Further, the likelihood ratio statistic $\lambda$ for $H$ is the product $\lambda = \lambda_1\lambda_2$ of $\lambda_2$ and the likelihood ratio statistic $\lambda_1$ for testing $H_1$ (Anderson [1], pp. 260-2). Unfortunately, both the distribution of $\lambda_1$ and the distribution of $\lambda_2$ depend upon the unknown $\Sigma$, and, unless $\Sigma$ is diagonal, $\lambda_1$ and $\lambda_2$ are dependent. As a result, this relationship between $\lambda$ and Bartlett's statistic is difficult to exploit in finding the distribution of $\lambda$. In this note, we use the invariance of $\lambda$ under orthogonal transformations to enable us to change to new variables having a diagonal covariance matrix. Homogeneity of variances for these new variables is shown to be equivalent to $H:\Sigma = \sigma^2I$ under the old variables. Using Anderson's representation for $H$, but now expressed in terms of the new variables, we have $\lambda = \lambda_1'\lambda_2'$, where $\lambda_2'$ tests homogeneity of variances for the new variables and $\lambda_1'$ tests the diagonality of the new covariance matrix. Since the new covariance matrix is diagonal, $\lambda_1'$ and $\lambda_2'$ are independent and $\lambda_1'$ has a distribution independent of the parameters. Such a representation is (hopefully) convenient for determining the properties of the likelihood ratio test based on $\lambda$. This representation, however, only connects the distribution of $\lambda$ and the distribution of Bartlett's statistic, for the "new" variables used in our representation are not observable, but rather are functions of the unknown covariance matrix $\Sigma$.

Let $X:p \times n$ be a matrix of random real variates such that the column vectors of $X$ are independently and identically distributed as multivariate normals with zero mean vectors. Then a positive definite quadratic function in normal vectors is defined as $XLX$' where $L$ is a symmetric positive definite (p.d.) matrix with real elements. In the analysis of variance, such functions appear. In the previous study, Khatri [14], [16], has established the necessary and sufficient conditions for the independence and the Wishartness of such functions. In this paper, we study the distribution of a positive definite quadratic function and the distribution of $Y' (XLX')^{-1}Y$ where $Y:p \times m$ is independently distributed of $X$ and its columns are independently and identically distributed as multivariate normals with zero mean vectors. Moreover, we study the distribution of the characteristic (ch.) roots of $(YY')(XLX')^{-1}$ and the similar related problems. When $p = 1$, the distribution of a p.d. quadratic function in normal variates (central or noncentral) has been studied by a number of people (see references). In the study of the above and related topics in multivariate distribution theory, we are using zonal polynomials. A. T. James [10], [11], [12], [13], and Constantine [1], [2], have used them successfully and have given the final results in a very compact form, using hypergeometric functions $_pF_q(S)$ in matrix arguments. These functions are defined by \begin{equation*}\tag{1}_pF_q(a_1, \cdots, a_p; b_1, \cdots, b_q; Z) \end{equation*} $= \sum^\infty_{k = 0} \sum_\kappa \lbrack (a_1)_\kappa \cdots (a_p)_\kappa/(b_1)_\kappa \cdots (b_q)_\kappa\rbrack\lbrack C_\kappa(Z)/k!\rbrack$ where $C_\kappa(Z)$ is a symmetric homogeneous polynomial of degree $k$ in the latent roots of $Z$, called zonal polynomials (for more detail study of zonal polynomials, see the references of A. T. James and Constantine), $\kappa = (k_1, \cdots, k_p), k_1 \geqq k_2 \geqq \cdots \geqq k_p \geqq 0, k_1 + k_2 + \cdots + k_p = k; a_1, \cdots, a_p, b_1, \cdots, b_q$ are real or complex constants, none of the $b_j$ is an integer or half integer $\leqq \frac{1}{2}(m - 1)$ (otherwise some of the denominators in (1) will vanish), \begin{equation*}\tag{2}(a)_\kappa = \prod^m_{j = 1} (a - \frac{1}{2}(j - 1))_{kj} = \Gamma_m(a,\kappa)/\Gamma_m(a), \end{equation*} (x)_n = x(x + 1) \cdots (x + n - 1), (x)_0 = 1$ and \begin{equation*}\tag{3}\Gamma_m(a) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a - \frac{1}{2}(j - 1)) \end{equation*} and $\Gamma_m(a, \kappa) = \pi^{\frac{1}{4}m(m - 1)} \prod^m_{j = 1} \Gamma(a + k_j - \frac{1}{2}(j - 1)).$$ In (1), $Z$ is a complex symmetric $m \times m$ matrix, and it is assumed that $p \leqq q + 1$, otherwise the series may converge for $Z = 0$. For $p = q + 1$, the series converge for $\|Z\| < 1$, where $\|Z\|$ denote the maximum of the absolute value of ch. roots of $Z$. For $p \leqq q$, the series converge for all $Z$. Similarly we define \begin{equation*}\tag{2b}_pF^{(m)}_q (a_1, a_2, \cdots, a_p; b_1, \cdots, b_q; S, R)\end{equation*} $ = \sum^\infty_{k = 0} \sum_\kappa\lbrack (a_1)_\kappa \cdots (a_p)_\kappa/(b_1)_\kappa \cdots (b_q)_\kappa\rbrack\lbrack C_\kappa(S)C_\kappa(R)/C_\kappa(I_m)k!\rbrack.$ The Section 2 gives some results on integration with the help of zonal polynomials, the Section 3 derives the distributions based on p.d. quadratic functions, the Section 4 gives the moments of certain statistics arising in the study of multivariate distributions, and the Section 5 gives the results for complex multivariate Gaussian variates.

A knockout tournament is a procedure for selecting the best among $2^n$ players by, in the first round, splitting the $2^n$ players into $2^{n - 1}$ pairs who play each other; the $2^{n - 1}$ winners proceed to the next round and repeat the process; finally the one player left is declared the best. A method is given for estimating a complete ranking of the $2^n$ players given the results of the $(2^n - 1)$ matches in the tournament; the method is based on the assumption that all $(2^n)$! orderings of the players are equally probable before the tournament begins.

In this note we give a result on the maximum sample excursions of Kiefer-Wolfowitz stochastic approximation processes. The method is applicable to other stochastic approximation procedures, and under other conditions than those assumed here. Let $y(x)$ be a scalar valued random variable with distribution function $H(y \mid x)$, where $x$ is a scalar valued parameter. Define $M(x) = \int yH(dy \mid x)$. Let $M(x)$ be continuous and have a unique local maximum at $x = \theta$ and let $a_n, c_n$ be sequences of positive real numbers satisfying \begin{equation*}\tag{1}\sum a_n = \infty,\quad\sum a_n^2c_n^{-2} < \infty,\quad\sum a_nc_n < \infty,\quad c_n \rightarrow 0.\end{equation*} The sequence of random variables $x_n$ defined by \begin{equation*}\tag{2}x_{n + 1} = x_n + a_n\lbrack y(x_n + c_n) - y(x_n - c_n)\rbrack/c_n,\quad x_0 \text{given},\end{equation*} is known as a Kiefer-Wolfowitz process and, under mild conditions on $M(x)$ and $H(y \mid x), x_n$ is known to converge to $\theta$ w.p.1. (See, e.g., Schmetterer [6] for a review of such results.) A result of this note is an estimate of the following form, for any $m < \infty$ and even integer $r$, \begin{equation*}\tag{3}P\lbrack\max_{m \geqq n \geqq N} |x_n - \theta| > \epsilon\rbrack < \lbrack E(x_N - \theta)^r + \delta_{Nr}\rbrack/\epsilon^r\end{equation*} where $\delta_{Nr}$ depends on the sequences $a_n$ and $c_n$ and can be made arbitrarily small for each fixed $N$ and $r$, while $x_n \rightarrow \theta$ w.p.1. is still insured. As a special case of (3), let $|x_N - \theta|$ be unknown but assumed nonrandom. Let $\epsilon = \beta + (1 + \alpha)|x_N - \theta|, \beta > 0$ and $\alpha > 0$. Then \begin{equation*}\tag{4}P\lbrack\max_{m \geqq n \geqq N} |x_n - \theta| > (1 + \alpha)|x_N - \theta| + \beta\rbrack \end{equation*} $< \lbrack(x_N - \theta)^r + \delta_{Nr}\rbrack/\lbrack \beta + (1 + \alpha)|x_N - \theta|\rbrack^r$ which can be made arbitrarily small by fixing $r$ sufficiently large, and then arranging $a_n$ and $c_n$ so that $\delta_{Nr}$ is sufficiently small. Aside from the intrinsic interest of (3) and (4), these results seem to have some practical usefulness in assisting in the choice of the $a_n$ and $c_n$ when there is more than one local maximum of $M(x)$, or if the process (2) is used to optimize the parameters of a physical system whose performance $(M(x))$ should not be reduced below some minimum level--the value of $x$ corresponding to this level may not be known. In both of these cases we may wish to limit the excursions to some given multiple or function of $|x_0 - \theta|$, with a high probability, while still being certain that $x_n \rightarrow \theta$ w.p.1.