Results of Chernoff-Savage (1958) and Govindarajulu-LeCam-Raghavachari (1966) are extended from the two-sample case to the general regression case and, simultaneously, the conditions on the scores-generating function are relaxed. The main results are stated in Section 2 and their proofs are given in Section 5. Sections 3 and 4 contain auxiliary propositions, on which the methods of the present paper are based. Section 6 includes a counterexample, showing that the theorem cannot be extended to discontinuous scores-generating functions.

We investigate the question of when a semi-Markov process is transformed by a random time change into a Markov process.

In this note, the notion of an optimal transform of a (discrete parameter) stochastic process is introduced. Such transforms are shown to exist in certain cases, and a relationship to optimal stopping times is discussed. These ideas lead naturally to the representation of any given stochastic process as the transform of a submartingale. This type of representation theorem is extended to continuous parameter processes, where it is shown that in certain cases a quasi-martingale can be represented as a stochastic integral with respect to a submartingale.

Let $\{Y_j\}^\infty_{j = 1}$ be a sequence of random variables defined on a probability space $(\Omega, F, P)$, which are not necessarily independent, or identically distributed. Let $S_n = Y_1 + \cdots + Y_n$. Assume that there exists sequences of constants $\{A_n\}, \{B_n\}, B_n > 0$ such that the limit distribution of $(S_n - A_n)/B_n$ exists. For a class of limit distributions which includes the stable distributions, we give a characterization of $\{B_n\}$ in terms of the dispersion of the sequence of partial sums $S_n$. Such a characterization will be useful in obtaining stable limit theorems for Markov chains since it allows a description of the norming constants which is not dependent on any particular state of the state space of the Markov chain. In addition, using this characterization and that of Tucker in [6], we obtain a partial strengthening of Paul Levy's Theorem on the Augmentation of the Dispersion.

This paper considers decision processes on a denumerable state space. At each state a finite number of decisions is allowed. The main assumption is that if one always chooses the same decision at each state the resulting Markov chain is ergodic (i.e. positive recurrent). Under this assumption it is shown that all possible decision procedures are (in an appropriate sense) uniformly ergodic.

Among various tests of independence the ones based on the sample correlation coefficient and on the $2 \times 2$ contingency tables seem to be foremost in applications. Although the first of these tests the absence of linear relation, the motivation stems from the fact that in the case where the bivariate distribution is a member of the normal family uncorrelatedness is equivalent to independence. A natural question arises, whether there exists a wider family of bivariate distributions where independence is characterized by uncorrelatedness. The answer to this question is given in a recent paper of Lehmann (1966). In the case of multivariate distributions the similar question is more involved since pairwise independence, in general, is not enough for mutual independence. In the present paper a simple generalization of the notion of uncorrelatedness is shown to characterize independence in a family of multivariate distributions which is analogous to the bivariate family considered by Lehmann (1966). When the data is available in the form of a $2 \times 2$ contingency table one might consider it as a simplified version of the data available on a pair of real random variables $(X_1, X_2)$ or that the information available to the experimenter is only in the form of occurrence or nonoccurrence of the events $X_1 \leqq a$ and $X_2 \leqq b$ where the pair $(a, b)$ is fixed. In both situations, one tests independence of the two events $\lbrack X_1 \leqq a\rbrack$ and $\lbrack X_2 \leqq b\rbrack$ although it may be desirable to test the independence of $X_1$ and $X_2$. Again, one might ask the question whether there exists a suitable family of bivariate distributions where the independence of the events of the above type characterizes the independence of the paired random variables. In the present paper such a family is given and a multivariate analogue of the same is shown to possess similar characterization of independence. In a recent paper Esary, Proschan and Walkup (1967) have introduced a notion of association which has several applications. Although disjoint from their study, the results of the present paper are in the same set-up and are supplementary. In order to give a precise summary, let $(X_1, X_2)$ be a pair of real valued random variables with finite second moments. The pair is said to be positively quadrant dependent if \begin{equation*}\tag{0.1}P\lbrack X_1 \leqq x_1, X_2 \leqq x_2\rbrack \geqq P\lbrack X_1 \leqq x_1\rbrack P\lbrack X_2 \leqq x_2\rbrack,\quad\text{for all} \quad x_1, x_2,\end{equation*} and negatively quadrant dependent if the inequality between the two sides of (0.1) is reversed. The bivariate distribution functions which satisfy the above restrictions define families $\mathscr{F}_1$ and $\mathscr{G}_1$ respectively. In a recent paper, Lehmann (1966) showed that in $\mathscr{F}_1 \mathbf{\cup} \mathscr{G}_1$, the independence is characterized by uncorrelatedness. In the same paper he also defined a subclass $\mathscr{F}_2 \subset \mathscr{F}_1$ which described the regression dependence between $X_1$ and $X_2$, and gave several applications to tests of the hypotheses of independence. Recently, Jogdeo and Patil (1967) showed that if $\mathscr{F}_2$ is parametrized suitably then the independence of $X_1, X_2$ is characterized by the independence of two events $\lbrack X_1 \leqq a\rbrack$ and $\lbrack X_2 \leqq b\rbrack$, for some $a$ and $b$, with the sole condition that the probabilities of these events be bounded away from 0 or 1. In particular, it was shown that if the dependence between $X_1$ and $X_2$ is described by a linear model \begin{equation*}\tag{0.2}X_1 = \alpha + \beta X_2 + \sigma Z,\end{equation*} where $X_2$ and $Z$ are independent, then the above characterization applies. In the present paper the parametrization is replaced by making the condition of regression dependence symmetric in both variables. In particular, the class $\mathscr{F}_3 \mathbf{\cup} \mathscr{G}_3$ discussed by Lehmann (1966) is a subclass of the one considered presently. In Section 2, the results stated above are generalized to multivariate distributions. Since it is well known that the pairwise independence is not enough for mutual independence, the conditions which characterize mutual independence take various forms and interpretations. The basic characteristic of the class of $n$-variate distributions which yields simple characterizations may be described as follows. If $A_i$ denotes the event $X_i \leqq x_i$ (or $\geqq x_i$) then $P(\mathbf{\cap} A_i)$ is either $\geqq$ or $\leqq \prod P(A_i)$ uniformly in $x_i$. For example, in the family of trivariate distributions which satisfy \begin{equation*}\tag{0.3}P\lbrack X_1 \leqq x_1, X_2 \leqq x_2, X_2 \leqq x_3\rbrack \leqq \prod^3_{i = 1}P\lbrack X_i \leqq x_i\rbrack\quad\text{for all}\quad x_1, x_2, x_3,\end{equation*} the independence is characterized by \begin{equation*}\begin{align*}\tag{0.4} EX_iX_j &= EX_iEX_j;\quad i \neq j, i = 1, 2, 3, \\ EX_1X_2X_3& = EX_1EX_2EX_3\end{align*}.\end{equation*} If the condition (0.3) is made stronger by requiring \begin{equation*}\tag{0.5}h(x_k; x_i, x_j) = P\lbrack X_i \leqq x_i, X_j \leqq x_j | X_k = x_k\rbrack,\quad i \neq j \neq k;\quad i = 1, 2, 3,\end{equation*} to be monotone in $x_k$ for every $x_i, x_j$ fixed then the independence of $X_1, X_2$ and $X_3$ is equivalent to that of the three events $\lbrack X_1 \leqq a\rbrack, \lbrack X_2 \leqq b\rbrack$ and $\lbrack X_3 \leqq c\rbrack$, for some $a, b, c$ such that the probabilities of these events are bounded away from 0 and 1.

In [4] we introduced a general method for obtaining asymptotically pointwise optimal procedures in sequential analysis when the cost of observation is constant. The validity of this method in both estimation and testing was established in [4] for Koopman-Darmois families, and in [5] for the general case. Section 2 of this paper generalizes Theorem 2.1 of [4] to cover essentially the case of estimation with variable cost of observation. In Section 3 we show that in estimation problems, under a very weak condition, for constant cost of observation, the asymptotically pointwise optimal rules we propose are optimal in the sense of Kiefer and Sacks [9]. The condition given is further investigated in the context of Bayesian sequential estimation in Section 4 and is shown to be satisfied if reasonable estimates based on the method of moments exist. In Section 5 we consider the robustness of our rules under a change of prior. The main result of this section is given by Theorem 5.1. Finally Theorem 5.2 deals with a generalization of Wald's [12] theory of asymptotically minimax rules and an application of that theory to the Bayesian model.

In a previous paper [Fabian (1967)], we have shown that the Kiefer-Wolfowitz procedure--for functions $f$ sufficiently smooth at $\theta$, the point of minimum--can be modified in such a way as to be almost as speedy as the original Robbins-Monro method. The modification consists of taking more observations at every step and utilizing these (according to a design $d$) so as to eliminate the effect of all derivatives $\partial^jf/\lbrack\partial x^{(i)}\rbrack^j, j = 3, 5, \cdots, s - 1$. Let $\delta_n$ be the distance of the approximating value to the approximated $\theta$ after $n$ observations taken. Under some regularity conditions it was shown that $E\delta^2_n = O(n^{-s/(s + 1)}).$ There are many designs $d$ achieving this speed. For selection of the best one, i.e. the one which minimizes $\lim n^{s/(s + 1)}E\delta^2_n$ we have to derive the dependence of this limit on the design $d$, which is done in Section 4. The best choice of the design $d = \lbrack u, \xi\rbrack$ is that which minimizes the right-hand side of (2.7) below; here $u = \lbrack u_1, u_2, \cdots, u_m\rbrack, \xi = \lbrack\xi_1, \cdots, \xi_m\rbrack$ with $0 < u_1 < u_2 < \cdots < u_m \leqq 1, \xi_i \geqq 0, \sum^m_{i = 1}\xi_i = 1; \xi_i$ indicates how many observations should be taken (roughly speaking) at $u_i$. The vector $v = \lbrack v_1, \cdots, v_m\rbrack$ is determined by $v = \frac{1}{2}U^{-1}e_1 (e_1 = \lbrack 1, 0, \cdots, 0\rbrack, \lbrack\cdots\rbrack$ denotes column vectors), $U^{(ij)} = u^{2i - 1}_j, i, j = 1, \cdots, m$. It seems difficult to minimize (2.7) given $K_0, K_1$. Moreover we usually do not know these constants. So in this paper we solve the question of minimizing the first term $\sum^m_{i = 1}(v^2_i/\xi_i)$ only. The result is formulated in Theorem 5.1.

The present paper proposes a definition of relative concentration of random variables about a given constant, and studies the relationship between two stochastic denominators $Z_1$ and $Z_2$ which causes the random quotient $X/Z_1$ to be more concentrated about zero than $X/Z_2$. In this paper we shall always assume that the numerator and denominator are independent. Two necessary and sufficient conditions, and several sufficient conditions for $X/Z_1$ to be more concentrated about zero than $X/Z_2$ are given in Section 4. The results of Section 4 are used in Section 5 to obtain generalizations of a theorem due to Hajek (1957) on the generalized Student's $t$-distribution. Sections 6 and 7 use these generalized theorems to produce tests and confidence intervals for several Behrens-Fisher type problems. Finally, Section 8 contains a proof of the randomization theorem stated in Section 4. In particular, Section 6 is concerned with the extension of a result in Lawton (1965) on Lord's $u$-statistic to the case of unequal sample sizes. Section 7 gives methods for constructing confidence intervals for linear combinations of means from $k$ normal populations

Let $N$ be a positive integer. In Section 2 an expository account in terms of moment space dependence is given of the Bayes estimate of a random probability $\Theta$, relative to squared difference loss, from an observable $X$ which given $\Theta$ is conditionally binomial $(N, \Theta)$. The risk and Bayes envelope functional are also considered in these terms. In Section 3 an explicit formulation is given for the minimax estimate of $\Theta$ when its first $N$ moments are known. Theorem 2 characterizes the condition that a Bayes estimate have constant risk over the class of all "priors" which yield these moments. In Section 4, a transformation is introduced which puts the interior of the space of the first $N$ moments for distributions on $\lbrack 0, 1\rbrack$ in one-one correspondence with the interior of the $N$-dimensional unit cube. This transformation is used to show that the supremum of the difference between minimax and Bayes risks over the class of all prior distributions is bounded above by $2^{-N}$. Examples for $N = 1, 2$, and 3 in terms of the above transformation are considered in Section 5.

Let the random variables $X_{i1}, X_{i2}, \cdots, X_{in}, i = 1, 2$, be real valued and independent with density functions $f(x - \theta_i) (\theta_i \text{real}), i = 1, 2$, (with respect to Lebesgue measure). We take $\int_{-\infty}^\infty xf(x) dx = 0$ with no loss of generality. The problem considered here is estimation of the function $\varphi(\theta_1, \theta_2) = \text{maximum} (\theta_1, \theta_2)$ with a squared error loss function. Questions of minimaxity and admissibility of certain natural estimators are considered. This problem is the estimation analogue of the well known ranking and selection problem, which has received considerable attention in the past. For a bibliography see Bechhofer, Kiefer, and Sobel (1968). Whereas previous work is concerned with choosing the population with the larger parameter, here we are concerned with estimating the larger parameter. Consider for the moment, the case where $n = 1$. A natural estimator of $\varphi(\theta_1, \theta_2)$ is $\varphi(X_{11}, X_{21}) = \text{maximum of} (X_{11}, X_{21})$. This estimator is symmetric in the observations and invariant under translations which take $(\theta_1, \theta_2)$ to $(\theta_1 + a, \theta_2 + a)$, for any real constant $a$. Under suitable conditions, one of which is that $f$ be symmetric, (the conditions will be stated precisely for general $n$ in the next paragraph), it is shown that $\varphi(X_{11}, X_{21})$ is minimax. However $\varphi(X_{11}, X_{21})$ is not in general admissible. Furthermore, when $f$ is not symmetric $\varphi(X_{11}, X_{21})$ also need not be minimax. Another estimate with the invariance properties stated above is the a posteriori expected value of $\varphi(\theta_1, \theta_2)$, given $(X_{11}, X_{21})$, when the generalized prior distribution of $(\theta_1, \theta_2)$ is taken to be the uniform distribution over the two dimensional plane. For many estimation problems, such an adaptation of the Pitman estimator is known to be minimax. However, for this problem, under suitable conditions (again including that $f$ be symmetric), this estimator need not be minimax. It is true though, that this estimator is admissible. In order to summarize the results for general $n$, it is convenient to define the analogues of the estimates considered above. Let $X_i$ be the a posteriori expected value of $\theta_i$, given $X_{ij}, j = 1, 2, \cdots, n$, when $\theta_i$ has the generalized uniform distribution as a priori distribution. That is, \begin{equation*}\tag{1.1}X_i = \int \theta_i \prod^n_{j = 1} f(X_{ij} - \theta_i) d\theta_i/\int \prod^n_{j = 1} f(X_{ij} - \theta_i) d\theta_i, i = 1, 2.\end{equation*} Note $X_i$ is the usual Pitman estimator of $\theta_i$ and thus if $f$ is the normal density then $X_i = \bar{X}_i$. Let \begin{equation*}\tag{1.2}\varphi(X_{11}, X_{12}, \cdots, X_{1n}, X_{21}, \cdots, X_{2n}) = \varphi(X_1, X_2) = \text{maximum of} (X_1, X_2).\end{equation*} Also define $\delta^\ast$ to be the a posteriori expected value of $\varphi(\theta_1, \theta_2)$, given $X_{ij}, i = 1, 2, j = 1, 2, \cdots, n$, when the generalized prior distribution of $(\theta_1, \theta_2)$ is the uniform distribution on the two dimensional plane. That is, \begin{equation*}\begin{align*}\tag{1.3}\delta^\ast(X_{11}, X_{12}, \cdots, X_{2n}) \\ &= \iint \varphi(\theta_1, \theta_2) \prod^n_{j = 1}f (X_{1j} - \theta_1) \prod^n_{j = 1}f(X_{2j} - \theta_2) d\theta_1 d\theta_2 \\ \cdot\lbrack\iint \prod^n_{j = 1}f(X_{1j} - \theta_1) \prod^n_{j = 1}f (X_{2j} - \theta_2) d\theta_1 d\theta_2\rbrack^{-1}\end{align*}. \end{equation*} In order to explicitly state the main results it is convenient to introduce some notation which will be more formally presented in the next section. Recall $X_i$ is the usual Pitman estimator of $\theta_i$. Let $Y_i = (Y_{i1}, Y_{i2}, \cdots, Y_{i, n - 1})$ where $Y_{ij} = X_{i, j + 1} - X_{i1}$. Let $p(x, y)$ be the conditional density of $X_i$ given $Y_i$ when $\theta_i = 0$. Now for the problem of estimating $\varphi(\theta_1, \theta_2)$ the main results are as follows: (a) If $p(x, y) = p(-x, y)$ and $EE\lbrack(X^2_1 + X^2_2) |Y_1, Y_2\rbrack < \infty$, then $\varphi(X_1, X_2)$ is minimax. The proof of this result is based on an idea of Farrell (1964). (b) If $p(x, y) \neq p(-x, y)$, then the problem is complicated and in fact an example is given showing that $\varphi(\cdot)$ is not minimax. (c) The estimator $\varphi(\cdot)$ need not be admissible. When $f$ is normal, it is easily shown, using a theorem and remark of Sacks (1963) that $\varphi(\cdot)$ is inadmissible (Section 4). (d) if $E\{E^2\lbrack(X^2_1 + X^2_2)\cdot|\log (X^2_1 + X^2_2)|^\beta|Y_1, Y_2\rbrack\} < \infty$ for some $\beta > 0$, then $\delta^\ast(\cdot)$ is admissible. The proof of this fact depends on results of Stein (1959b) and James and Stein (1961). (e) The estimator $\delta^\ast$ need not be minimax and in Section 3 it is shown that frequently $\delta^\ast$ is not minimax. Despite the inadmissibility of $\varphi(X_1, X_2)$, this estimator has great intuitive appeal, is easy to use, and we have not succeeded in finding an estimate whose risk improves on the risk of $\varphi$. Also it is minimax under the restrictions noted above. Furthermore, if the problem was to simultaneously decide which population has the larger mean and estimate the larger mean, then a reasonable formulation for which $\varphi(X_1, X_2)$ is an admissible estimator of $\varphi(\theta_1, \theta_2)$ could easily be found. See for example Cohen (1965). The admissible $\delta^\ast$, on the other hand, is more difficult to compute and has a larger bias than $\varphi(X_1, X_2)$. In a future paper, we shall discuss topics such as the general question of unbiased estimation of $\varphi(\theta_1, \theta_2)$, the maximum likelihood estimate and its properties, and the generation of invariant Bayes estimates of which $\delta^\ast$ is a particular example. As we develop the main results, generalizations will be indicated. In the next section we introduce the notation for a more general model than as given earlier and develop some preliminaries. In Section 3 we give the minimax results while in Section 4 we give the admissibility results.

Let the random variables $X_{i1}, X_{i2}, \cdots, X_{in}, i = 1, 2$, be real valued and independent with density functions $f(x - \theta_i) (\theta_i \text{real}), i = 1, 2$, (with respect to Lebesgue measure). We take $\int^\infty_{-\infty} xf(x) dx = 0$ with no loss of generality. The problem is to estimate the ordered pair $(\theta_1, \theta_2)$, under the condition $\theta_2 \geqq \theta_1$, when the loss function is the sum of the squared errors in estimating the individual components. Questions of minimaxity and admissibility of the analogue of the Pitman estimator are considered. This problem, which represents a two dimensional estimation problem subject to constraints, has received attention in the past. Most of the literature deals with obtaining maximum likelihood estimates for specified densities. (See for example Brunk [3].) Katz [6] considers some aspects of the problem for the binomial and normal densities. The analogue of the Pitman estimator studied here is the vector estimator which is the a posteriori expected value of $(\theta_1, \theta_2)$ given $X_{ij}, i = 1, 2, j = 1, 2, \cdots, n$, when the generalized prior distribution is the uniform distribution on the half space $\theta_2 \geqq \theta_1$. If we call this estimator $\delta = (\delta_1, \delta_2)$, then \begin{equation*}\begin{align*}\tag{1.1}\delta_i(X_{11}, X_{12}, \cdots, X_{2n}) \\ = \iint_{\theta_2 \geqq \theta_1} \theta_i \prod^n_{j = 1} f(X_{1j} - \theta_1) \prod^n_{j = 1}f(X_{2j} - \theta_2) d\theta_1 d\theta_2 \\ \cdot\lbrack\iint_{\theta_2 \geqq \theta_1} \prod^n_{j = 1}f(X_{1j} - \theta_1) \prod^n_{j = 1}f(X_{2j} - \theta_2) d\theta_1 d\theta_2\rbrack^{-1},\quad i = 1, 2\end{align*}.\end{equation*} In order to state the main results, it is convenient to introduce some notation. Let \begin{equation*}\tag{1.2}X_i = \int \theta_i \prod^n_{j = 1}f(X_{ij} - \theta_i) d\theta_i/\int \prod^n_{j = 1}f(X_{ij} - \theta_i) d\theta_i,\quad i = 1, 2.\end{equation*} Let $Y_i = (Y_{i1}, Y_{i2}, \cdots, Y_{i, n - 1})$ where $Y_{ij} = X_{i,j + 1} - X_{i1}$. Let $p(x, y)$ be the conditional density of $X_i$ given $Y_i$ when $\theta_i = 0$. We obtain the following results: (a) If $EE\lbrack(X^2_1 + X^2_2)\mid Y_1, Y_2\rbrack < \infty$, and if $p(x, y) = p(-x, y)$ then the Pitman estimator $(\delta_1, \delta_2)$ given in (1.1) is minimax. The normal and uniform densities are examples of when this condition is satisfied. (b) Let $P(x, y)$ denote the cumulative distribution function corresponding to $p(x, y)$. That is, for a fixed $y$, $P(x, y) = \int^x_{-\infty}p(u, y) du.\quad\text{If}\quad EE\lbrack(X^2_1 + X^2_2)\mid Y_1, Y_2\rbrack < \infty$ and if $p(x, y)$ is such that for each $y, p(x, y)/(1 - P(x, y))$ increases in $x$, (i.e. increasing hazard rate) and $p(x, y)/P(x, y)$ decreases in $x$, then the Pitman estimator is minimax. The family of gamma densities, $f(t) = t^\alpha \exp (-t)/\Gamma(\alpha + 1)$ for $\alpha > 0$, are examples of when the above condition is satisfied. The proofs of the minimax results (a) and (b) are based essentially on the method of Farrell [4]. (c) An example is given which indicates that in general the Pitman estimator is not minimax. The example is justified by a computation performed by numerical integration. The numerical integration shows that the risk of the Pitman estimator exceeds the risk of an estimator known to be minimax. The results (a) (b), and (c) indicate that whereas in a related one dimensional problem, namely to estimate a translation parameter $\theta$, subject to $\theta \geqq 0$, (see Farrell [4], Section 7), the Pitman estimator is always minimax (save for moment and continuity conditions), the same is not true for this two dimensional problem. (d) Let \begin{equation*}\tag{1.3}\rho(y) = \max \{\sup_{-\infty < x < -1} \lbrack\int^\infty_{-\infty} v dv \int^\infty_{-\infty} p(u - v, y_1) \cdot p(u + v, y_2) du/x \int^x_{-\infty} \int^\infty_{-\infty} p(u - v, y_1)p(u + v, y_2) du dv\rbrack, 2\}.\end{equation*} If \begin{equation*}\tag{1.4}E\rho^2(y)E\lbrack(X^4_1 + X^4_2) \cdot(1 + |\log (X^2_1 + X^2_2)|^\beta|Y_1, Y_2\rbrack < \infty,\quad\text{for some} \beta > 0,\end{equation*} then the Pitman estimator given in (1.1) is admissible. The normal density is an example for which (1.4) holds. Whereas Katz [6] stated the admissibility result for the normal case, the proof there is not adequate. The proof given here is based on results of Stein [9], and James and Stein [5]. In the next section we give notation. The minimax results are given in Section 3, and admissibility in Section 4.

The paper studies hypothesis testing problems $(H, K_1)$ for the mean of a vector variate having a multivariate normal distribution with known covariance matrix, in cases where the alternative $K_1$ is restricted by a number of linear inequalities (Section 2). We describe a method for obtaining the most stringent size-$\alpha$ test $\varphi^\ast$ for Problem $(H, K_1)$ (Section 3). This method is used for constructing $\varphi^\ast$ for a number of special problems $(H, K_1)$ (Sections 5, 6 and 7) where (i) $K'_1$ is symmetrical and (ii) $\psi_0$ is sufficiently small. Thus for these special problems, we can compare $\varphi^\ast$ with the most stringent SMP size-$\alpha$ test $\varphi_0$ that can be obtained by applying the general methods of [9] and [10] (described in Section 2). It turns out (for the special problems with $\alpha = .05$ or .01) that $\varphi^\ast$ does not provide a worth-while improvement upon $\varphi_0 : \varphi^\ast$ has a smaller maximum shortcoming but $\varphi_0$ is better than $\varphi^\ast$ from other over-all points of view. This supports the opinion that generally no serious objections can be made to the use of the MSSMP tests constructed in [10] Part 2 for problems from actual practice. This is a fortunate circumstance, for these MSSMP tests require only simple calculations. By the way we prove a theorem characterizing the MS size-$\alpha$ test for a very general problem $(H, K_{(m)})$ where the hypothesis $H$ may be composite while $K_{(m)})$ consists of a finite number $(m)$ of parameters $\theta_i(i = 1,\cdots, m)$ (Section 4).

Whittle's (1957), (1958) smoothed probability-mass-function estimate, determined by a Bayesian criterion, depends on a choice of the mean and covariance structure of a prior distribution on the unknown probabilities. A data-analytic choice is proposed (Equations (7) and (8)), based on natural stationary assumptions (Equations (4)). Adaptations and modifications for multidimensional cells are given (Section 4) and some alternatives are mentioned (Section 5).

A parametric probability model for order data on a set of objects is developed, which is appropriate for analysis of data with tree structure; the results of knockout tournaments are an example of such data. The parameters of the model are the probabilities that the various objects will be ranked highest. An exact Bayes analysis is possible if the prior distribution of the parameters is Dirichlet. The 1965 Wimbledon tennis tournament is used as an example.

The response variable of a general multivariate model can be constructed as a positive affine transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the multivariate model has parameters that can be expressed as the mean vector and the variance matrix. In the case of an error variables that is standard normal, it becomes the ordinary multivariate normal. For the general multivariate model with rotationally-symmetric error the sample inner-product matrix is a conventional statistic for inference. The distribution of this statistics is derived: the distribution is a conditional distribution given observable characteristics of the error variable. The response variable of a more specialized multivariable model can be constructed as a positive linear transformation of a vector error variable. In the case of an error variable that is rotationally symmetric, the model has parameters that can be expressed in terms of the variance matrix, the mean vector being zero. In the case of an error variable that is standard normal; it becomes the ordinary central multivariate normal. For the multivariate model with rotationally-symmetric error, the distribution of the Wishart matrix is derived; the distribution is a conditional distribution given observable characteristics of the error variable. And for the multivariate model with standard normal error, the noncentral distribution of the Wishart matrix is also derived, again as the appropriate conditional distribution.

The polykays of Tukey [8] and the bipolykays of Hooke [7] were generalized by dayhoff [4] for arbitrary balanced complete finite population structures. The expected mean squares in the analyses of variance of such structures may by expressed as linear function of variance components corresponding to the factors classifying the population. Since the variance components serve to measure the relative influence of the factors it is often desired to estimate these quantities. Unbiased estimates may be obtained by substituting observed mean squares for population mean squares and solving the resulting linear equations for the variance components. Alternative expressions for the expected mean squares involve linear functions of quantities called capsigmas (Wilk [10], Zyskind [11], White [9]), and the variance component estimates may be expressed as linear functions of sample cap sigmas. Dayhoff [3] shows that cap sigmas are generalized polykays of degree two and that the variances and covariances of variance components are linear functions of generalized polykays of degree four. Since generalized polykays have the property of inherence on the average (i.e., averages of sample generalized polykays over all random samples are the same generalized polykays of populations responses, the variances and covariances of the unbiased variance component estimates may be estimated unbiasedly by linear functions of generalized polykays of degree four. The generalized polykays are in turn linear functions of generalized symmetric means (Hooke [6], Dayhoff [4]) which may be computed directly from the observations. However, such computations are very difficult to carry out by hand because the formulas may involve thousands of distinct generalized polykays and generalized symmetric means for moderately large numbers of levels of the factors is not only impossible to carry out by hand but even for too costly using the most advanced digital computers, because a single generalized symmetric mean may require hundreds of billion of operations in its evaluation [2]. In order to make the computations of the variance-covariance matrix for estimated variance components economically feasible several requirements must be met. First the variance-covariance formulas in terms of the generalized symmetric functions must be generated by computer algorithms. Secondly, a way of computing the generalized symmetric means must be developed which significantly reduced the number of additions and multiplications. Thirdly, the above algorithms must be made general enough for application to the many possible balanced complete response structures which may be encountered as the relationship of nesting among the factors varies. In meeting the above requirements it is necessary to determine a logical system of relationships which allows the development of a few relatively simple algorithms to perform the various tasks on the computer, and which may be applied generally to the many different structures possible. The present paper results from an attempt to find such logical relationships among the various quantities which may be used in programs for digital computation of the variance-covariance matrix of estimated variance components. Study of the patterns of subscript restrictions which specify the generalized symmetric means leads to the development of algebraic relationships between the generalized polykys and the generalized symmetric means, which may be formulated in terms of a lattice of ordered partitions. Similar relationships exist between the numerators of the generalized symmetric means and quantities called unrestricted sums. These latter quantities may be computed much more efficiently than the generalized symmetric means themselves. (Hooke [6] gives an example of such a computation for a two factor structure using quantities similar to the unrestricted sums.) The various relationships, in addition to their intrinsic theoretical interest provide the necessary logical basis for the development of digital computer algorithms for performance of the algebraic and numerical computations for estimation of the variances and covariances of the variance component estimates.

In a recent paper [1], the authors presented a procedure showing how the treatment design matrix for an irregular fractional replicate of an $N$ treatment factorial might be adjusted to furnish estimates of the effect parameters as orthogonal linear functions of the observations.The procedure may be summarizes as follows: If the $N \times N$ design matrix $X$ of the complete factorial is partitioned into four submatrices $X{11}, X_{12}, X_{22}$ having dimension $(p + m_1) \times p, (p + m_1) \times m, m_2 \times p$ and $m_2 \times m$ respectively, with $m_1 + m_2 = m$ and $p + m = N, X_{11}$ would correspond to the design matrix of an irregular fractional replicate consisting of $p$ effect parameters. With the help of an auxiliary design matrix $\lambda$, the design matrix $X_{11}$ and the corresponding observation vector $Y_{p+m_1}$ of $(p + m_1)$ components were then augmented to become $X_1 = \lbrack X'_{11} : X'_{11} \lambda\rbrack'$ of dimensions $(p + m_1 + m_2) \times p$, and $Y_1 = \lbrack Y' : Y' \lambda\rbrack'$ of dimensions $(p + m_1 + m_2) \times 1$ in such a way that $\lbrack X_1'X_1\rbrack$ reduced to a diagonal matrix. The success of the procedure depended on being able to find least squares estimates for each of $m_2$ omitted observations, and this, in turn, was possible because it was possible to have least squares estimates for each of the effect parameters retained. In other words, this meant that both $(X'_{11} X_{11})^{-1}$ and $(X_{22}X'_{22})^{-1}$ existed. The structural relationship between the effect parameters retained and the observations omitted was such that existence of one of the inverses implied the existence of the other. If the rank of $(X'_{11}S_{11})$ is not full, either as a result of defective or intentional construction of the fractional replicate, the rank of $(X_{22}X'_{22})$ will be less than full, and, as a result, it will not be possible to have unique least squares estimates for each of the omitted observations. When this is the case, the corresponding fractional replicate would be what might be characterized as a singular fractional replicate. A question then arises as to how the results presented in [1] would be affected by this singuarlity. The question has been resolved and analogous results have been obtained through use of generalized inverses (referred to as $g$-inverses) in the present paper. By way of an aid to the derivation of the analogues, a few additional results on the ranks of the associated submatrices have also been obtained.

A method is given which can be used to construct procedures for discriminating among distributions on a Euclidean space with continuous distribution functions. The decision space used includes "partial" decisions and the probabilities of errors are random variables with beta distributions. Emphasis is upon control of the distribution of the conditional overall probabilities of errors. These procedures can be used in a wide class of discrimination problems, such as, for example, discriminating among multivariate normal distributions with unknown, unequal dispersion matrices. A number of other writers have suggested nonparameteric discrimination procedures. The first work in this area, to the knowledge of these writers, was by Fix and Hodges [2]. Since the work by those authors has remained unpublished, a brief statement of its approach and results is given in Section 4 for comparison. Procedures have been suggested also by Stoller [3], Anderson [1] and Kendall [8].

This note used the terminology and notation of Ross [3]. The example presented here is a denumerable Markovian decision process which has an optimal nonstationary rule which is better than every stationary rule. This answers a question of Derman [1].

Vartak [4] has shown by enumeration that the Knoecker product of two PBIB designs with $s$ and $t$ associate classes is again a PBIB with at most $s + t + st$ associated classes. In this paper the same result is established more easily with the help of association matrices. In addition to this, it is shown that the association matrices of a PBIB which is the Kronecker product of two known designs, are the Kronecker product of those of the original designs and that the "augmented matrices of the parameters of the second kind" of the resulting design are the Kronecker product of the corresponding matrices of the given designs.