The problem of finding Bayes solutions for sequential experimental design problems motivates the study of the following type of one-person sequential game. If the game is stopped at any stage $a$, the loss to the player is the value of a random variable (rv), say $Z_a$. If the player chooses to continue the game, he can select the next rv to be observed from a class of rv's available at that stage, thus bringing the game to one of the stages succeeding stage $a$. (The class of all stages can be pictured as a "tree".) At this stage the player can again choose to stop, accepting the value of the chosen rv as his loss, or he may continue by selecting one of the class of rv's now available for the next observation. The player is required to stop sometime, and his decisions at any stage must depend only on information available at that stage. A model for this situation is given in Section 4. Control variables, which correspond to stopping variables in the usual formulation of sequential games, are defined which can be used by the player to decide whether to stop or not at any stage and, if he continues, which rv to observe next. A general characterization of control variables that minimize expected loss is given, and existence of such optimal control variables is proved under conditions applicable to statistical problems. The application to finding Bayes solutions to sequential experimental design problems is given in Section 5. As a preliminary to the discussion on control variables, Section 3 provides a study of the theory of optimal stopping variables. Let $\{Z_n, F_n, n \geqq 1\}$ be a stochastic process on a probability space $(\Omega, F, P)$ with points $\omega$. A stopping variable (sv) is a rv $t$ with values in $\{1, 2, \cdots, \infty\}$ such that $t < \infty$ a.e. and $\{t = n\} \varepsilon F_n$ for each $n$. For any such sv $t$, a rv $Z_t$ is defined by \begin{align*}Z_t(\omega) &= Z_n(\omega),\quad\text{if} t(\omega) = n, \\ &= \infty,\quad\text{if} t(\omega) = \infty.\end{align*} It is convenient to think of $Z_n$ as the loss after $n$ plays in a one-person sequential game and to consider the $\sigma$-field $F_n$ as representing the knowledge of the past after $n$ plays. The problem of finding a strategy for stopping the game to minimize the expected loss corresponds to finding a minimizing sv, i.e., one which minimizes $EZ_t$ among the class of all sv's $t$. The main results in Section 3 are new characterizations of Snell's solution in [12] to the problem of optimal stopping which generalized the well-known Arrow-Blackwell-Girshick theory in [1]. Under the assumption that there is an integrable rv $U$ such that $Z_n \geqq U$ a.e. for each $n$, Snell showed that when a minimizing sv $t^\ast$ exists, it can be defined as the first integer $j$ such that $X_j = Z_j$ (or $\infty$ if no such integer exists) where $\{X_n, F_n, n \geqq 1\}$ is the maximal regular generalized submartingale relative to $\{Z_n, F_n, n \geqq 1\}$. Under the additional assumption that $\{Z_n, n \geqq 1\}$ is an integrable sequence, we now show that the rv's $X_n$ can be further identified as $X_n = \operatorname{ess} \inf_{t \varepsilon T_n} E(Z_t \mid F_n) \text{a.e.},$ where $T_n$ is the class of sv's $t$ such that $t \geqq n$. It follows that if there is a set $A_n$ in $F_n$ for each $n$ such that (i) $Z_n \leqq E(Z_t \mid F_n)$ a.e. on $A_n$ for each $t$ in $T_{n + 1}$, and (ii) $Z_n > \operatorname{ess} \inf_{t \varepsilon T_{n + 1}} E(Z_t \mid F_n)$ a.e. on the complement $A'_n$, then the minimizing sv $t^\ast$ defined above is such that, for almost all points $\omega, t^\ast(\omega) = n$ if and only if $\omega \varepsilon A_n - \mathbf\bigcup^{n - 1}_{i = 1} A_i$. This comes very close to stating that the player of the sequential game above should stop playing after $n$ plays if and only if there is no continuation (i.e., sv in $T_{n + 1}$) having conditional expected loss given the past which is less than the present loss $Z_n(\omega)$. Unfortunately, such an interpretation is not valid in general since it is equivalent to stopping the game after $n$ plays if and only if $Z_n(\omega) \leqq \inf_{t \varepsilon T_{n + 1}} E(Z_t \mid F_n)(\omega).$ The function $\inf_{t \varepsilon T_{n + 1}} E(Z_t \mid F_n)$ not only may not be measurable, but also in many cases it may be changed almost at will by choosing different versions of the conditional expectations involved. On the other hand, the interpretation above is valid, as is shown in Section 3, for the case where each $\sigma$-field $F_n$ is generated by finitely many discrete rv's. Section 3 also contains a reasonably general development of the theory of minimizing sv's, including a more direct proof of Snell's result using the definition $X_n = \operatorname{ess} \inf_{t \varepsilon T_n} E(Z_t \mid F_n)$. Illustrations are given and comparisons are made with the approaches used by Arrow, Blackwell, and Girshick in [1] and Chow and Robbins in [4] and [5].

This paper considers the following problem. One takes independent and identically distributed observations from a population obeying a probability law $F_\theta(x)$. However, one does not know $F_\theta(x)$. What is known is a family of distribution functions $\Theta = \{F_\theta(x)\}$ and one assumes that there exists a prior probability measure $\mu(d\theta)$ on $\Theta$. At each stage $n = 1, 2, \cdots$ of the sampling one may stop. If one does stop he gets a payoff: $m_n - nC = Y_n$, where $m_n = \text{maximum} (X_1, \cdots, X_n)$ and $C > 0$. One is interested in a procedure which is optimal. In Section 2 we define "procedure" and "optimal procedure." We show that under some conditions an "optimal procedure" exists. The optimal procedure turns out to be: Stop at stage $j$ if $Y_j = \alpha(j)$ where $\alpha(j)$ is a function of $X_1, X_2, \cdots, X_j\mu(d\theta)$. The function $\alpha(j)$, although easy to describe, is quite difficult to calculate, hence we give in Section 3 two other functions which again are functions of $X_1, \cdots, X_j, \mu(d\theta)$ and are somewhat easier to calculate. These functions, denoted by $\beta_1(j), \beta_2(j)$ have the following properties: $\beta_1(j) \leqq \alpha(j) \leqq \beta_2(j)$ and $\beta_2(j) - \beta_1(j) \rightarrow 0$ almost surely as $j \rightarrow \infty$. Furthermore, we give an example for which $Y_j < \alpha(j) \Leftrightarrow Y_j < \beta_1(j) \Leftrightarrow Y_j < \beta_2(j)$. The case for which $\mu(d\theta)$ is degenerate, namely the case for which $F_\theta(x)$ is known to the sampler was solved in [2]. The work done in [1], [2], [3] helped us to obtain the results in Sections 2 and 3. The work done in [4] and [5] deals with similar problems but the approach is different. All relations in this work are understood to hold almost surely unless otherwise specified.

The large sample behavior of posterior distributions is examined without the assumption that the model is correct. Under certain conditions it is shown that asymptotically, the posterior distribution for a parameter $\theta$ is confined to a set (called the asymptotic carrier) which may, in general, contain more than one point. The asymptotic carrier depends on the model, the carrier of the prior distribution and the actual distribution of the observations. An example shows that, in general, there need be no convergence (in any sense) of the posterior distribution to a limiting distribution over the asymptotic carrier. This is in contrast to the (known) asymptotic behavior when the model is correct; see e.g. [7], p. 304: the asymptotic carrier then contains only one point, the "true value" of $\theta$ and the posterior distribution converges in distribution to the distribution degenerate at the "true value."

Let $T$ be a compact metric space and let $D = \{t_1, t_2, \cdots\}$ be a countable dense subset. We propose to show that if for $x \varepsilon C(T)$ we define $x_n(t) = E(x(t) \mid x(t_1), \cdots, x(t_n))$ (conditional expectation) then $E\|x_n - x\|^p \rightarrow 0$ where $\|\cdot\|$ denotes the $\sup$ norm and $p \geqq 1$ is such that $E\|x\|^p < \infty$. Furthermore, if we measure the distance between $x_n$ and $x$ by $\int |x_n(t) - x(t)\|^2 d\mu(t)$ for some finite measure $\mu$ on $T$, then $x_n$ is (in a least square sense) the optimal prediction for $x$ given $x(t_1), \cdots, x(t_n)$. We also consider an optimization problem in this same probabilistic setting. Roughly stated, we consider how, given $x(t_1), \cdots, x(t_n)$, one should choose $t \varepsilon T$ so as to maximize $E(x(t))$. The existence of an optimal policy is proved. If we let $S(x)$ denote the supremum of $x$ over $T$ and let $v_n(x)$ denote $x(t)$ where $t$ is the point chosen in accordance with the optimal policy then it is shown that $E\|S(x) - v_n(x)\| \rightarrow 0$ as $n \rightarrow \infty$. This last result is obtained under the assumption that $E\|x\| < \infty$.

The object of this paper is the development of a theory of optimal one-sample goodness-of-fit tests and of optimal two-sample randomized distribution-free (DF) statistics analogous to the well-known results of Hoeffding (1951), Terry (1952), Lehmann (1953), (1959), Chernoff and Savage (1958), Capon (1961) and others for two-sample nonrandomized rank statistics. For $Y_1, \cdots, Y_n$ a random sample from a population with continuous distribution function $G$, one tests in the one-sample case $H_0 : G = F$ vs. $H_1 : G \neq F$, where $F$ is some known continuous distribution function. From the Neyman-Pearson lemma, distribution-free tests that are most powerful (MP) for any $H$ vs. $K$ satisfying $KH^{-1} = GF^{-1}$, are obtained. From these MP distribution-free tests, one can on paralleling the derivations ([14], [25], [17], [18], [7]) for locally MP tests in the two-sample case obtain locally MP tests in the one-sample case. Further, it is found that the class of alternatives, for which a critical region of the form $\lbrack\sum J\lbrack F(y_i)\rbrack > c\rbrack$ is locally MP, is the class of $G$'s that consists of "contaminated" Koopman-Pitman distributions as given in Section 5. Randomized versions of the two-sample MP and locally MP rank statistics are considered and shown to be asymptotically equivalent to the locally MP rank statistics.

Some of the concepts and results of Chapman [3] for one-sample distribution-free tests are extended to the two-sample problem. Chapman's lead will be followed since his work gives a goodness criterion for comparing distribution-free tests, for finite sample sizes, over a large class of one-sided alternatives. Since all two-sample distribution-free statistics known to the authors satisfy Scheffe's [10] boundary condition, and all, except those of Pitman, also are strongly distribution-free (SDF) and therefore [1] are rank statistics, consideration may reasonably be restricted to rank tests. Such tests with the additional property of monotonicity are unbiased, partially ordered, and assume maximum and minimum powers for certain reasonable alternatives. Some of the maximum powers of the Mann-Whitney-Wilcoxon, Fisher-Yates, van der Waerden, Doksum, Savage, Epstein-Rosenbaum, and Cramer-von Mises statistics are tabulated. (For definitions and references, see Section 6.)

Let $\mathscr{F} = \{F(x; y), y \varepsilon E\}$ be a family of cumulative distribution functions (cdf's) in the variable $x$ indexed by $y \varepsilon E$, where $E$ is a measurable subset of the real line. Assume that $F(x; y)$ is measurable in $y$ for all $x$. Then, for any nondegenerate cdf $G$, whose induced probability measure assigns measure one to $E$, the cdf $H_G(x) = \int_E F(x; y) dG(y)$ is called a "mixture" of $\mathscr{F}$, and $G$ is called the "mixing distribution." The family $\mathscr{F}$ will be called the kernel of the mixture. Among the many interesting problems engendered by mixtures of distributions is one of great practical importance, namely, the estimation of the mixing distribution. This problem is simply stated--on the basis of observations from the mixture $H_G$, estimate the mixing distribution $G$. However, before actual estimation can be meaningfully investigated, identifiability of the family of mixtures, defined presently, must be verified. Let $\mathscr{G}$ denote the class of mixing distributions $G$, and $\mathscr{H}$ the induced class of mixtures $H$ for some specified family $\mathscr{F}$. A mixture of $\mathscr{F}$ will be called "identifiable" if $H(x) = \int F(x; y) dG^\ast(y) = \int F(x; y) dG(y)$ implies $G^\ast = G$. If every member $H$ of $\mathscr{H}$ is identifiable, then $\mathscr{H}$ will be said to be identifiable. For identifiable families $\mathscr{H}$, the problem of estimation of the mixing cdf when the elements of $\mathscr{F}$ occurring in the mixture are known is dealt with here. In particular, the problem of unbiased estimation for finite mixtures is considered. In a finite mixture the kernel is any finite set of known but arbitrary cdf's and the mixing distributions are discrete assigning positive weight to each of the cdf's in the kernel. The estimation of the mixing ratio when two arbitrary known cdf's are mixed is discussed at length in Section 2. Identifiability is then evident. Necessary and sufficient conditions on two distribution mixtures for uniform attainment of the Cramer-Rao lower bound are derived. The class of $\theta$-efficient estimators is found; also, as it turns out, the minimax unbiased estimator is a member of this $\theta$-efficient class and it is characterized. In Section 3, the results of Section 2 are extended to any finite mixture. For identifiable finite mixtures, necessary and sufficient conditions for the existence of an estimator which uniformly attains the minimal ellipsoid of concentration are given. The $\theta$-efficient family of estimators is derived; also, estimators within the $\theta$-efficient family which are consistent asymptotically normal efficient are characterized.

In many multi-decision problems the existence of admissible decision functions (for definitions we refer to [5]) depends upon the existence of corresponding partitions of the sample space into regions of specified shape. Usually the requirements for such statistical partitions differ from those relating to less restricted partitions of a set $S$ according to a vector of finite measures on the measurable subsets of $S$, usually referred to under the general title of the "ham sandwich problem" (see, e.g., [3], where further references may be found). Nevertheless, as indicated here, the solutions to a wide class of division problems rest very heavily on the fundamental result of Lyapunov and generalizations of this (see, e.g., [4]). Several multi-decision problems (Section 4 below) relating to the mean $\mu$ of a $k$-variate normal distribution $N(\mu, \Sigma)$ reduce to the problem of locating (hence called "topothetical," cf. [2]) the parameter point $\mu$ into one of $k + 1$ convex $k$-dimensional polyhedral cones $\omega_1, \omega_2, \cdots, \omega_{k+1}$ (hereafter referred to as "cones") with common vertex $\mu_0$ which form a partition of the parameter space $E_k$ of $\mu$ (see (1) below). Let us identify the sample space $E_k$ of an observation $X$ from $N(\mu, \Sigma)$ with the parameter space $E_k$. It was shown in [2] that the family $R_\omega$ of all translations $R(\tau) = (R_1(\tau), \cdots, R_{k+1}(\tau))$ of the system $\omega = (\omega_1, \cdots, \omega_{k+1})$ (see Definition 2) defines a class of admissible procedures, henceforth referred to as partitions; the decision $d_i$ that $\mu \varepsilon \omega_i$ is taken when the actual observation $x \varepsilon R_i$. Furthermore, there exists a unique partition $R(\tau_0) \varepsilon R_\omega$ which is minimax. The minimax character of $R(\tau_0)$ amounts to the following proposition: There exists a unique, in $R_\omega$, partition of $E_k$ into $k + 1$ cones with the same probability content under the normal $k$-variate distribution (Corollary 5.1 of [2]). The distribution may be assumed spherical normal (unit variance in any direction) without any loss of generality, since a nonsingular linear transformation $T$ such that $T\Sigma T' = I$ preserves the shape of the partition $\omega$. The purpose of this note is to extend the above result to the case of arbitrary probability contents for such conical regions (Theorem 1), and at the same time show how the corresponding partitions are related to classes of admissible partitions for a family of classification and topothetical problems relating to the normal mean $\mu$. Several problems which have been extensively studied in the statistical literature emerge as special cases of our general topothetical problem (Section 4).

For a given family of distributions let $\varphi$ be a parameter which takes on different values on different subfamilies but is constant on each. The values of the parameter may be taken to define an ordering of the subfamilies. Various concepts of statistical reproduction of such an ordering are introduced. They generalize concepts such as unbiased estimators and tests. In the latter part of the paper it is assumed that in each subfamily all distributions may be obtained from a single one by translation of the random variable. As an application we consider testing for decrease in variability when the individuals tested may have different means.

The $T_0^2$-statistic was introduced by Hotelling [5], [6] as a measure of multivariate dispersion in connection with the problem of testing the accuracy of bombsights. A similar statistic was considered earlier by Lawley [10] as a generalisation of the $F$-test for testing significance in multivariate analyses of variance. In general, if the $m \times m$ matrices $S_1$ and $S_2$ are independently distributed on $n_1$ and $n_2$ degrees of freedom respectively, estimating the same covariance matrix, then $T_0^2$ is defined by $T = T_0^2/n_2 = \mathrm{tr} S_1S^{-1}_2.$ For instance, in a one-way classification analysis of variance $S_1$ and $S_2$ may be the "between classes" and "within classes" matrices of sums of squares and products. The matrix $S_1$ may be singular, i.e., $n_1 < m$, but $S_2$ is assumed non-singular. Assuming that one is sampling from a normal population, $S_2$ has the Wishart distribution and $S_1$ the (possibly) non-central Wishart distribution if $n_1 \geqq m$. If $n_1 < m$, the distribution of $T$ may be obtained from that for $n_1 \geqq m$ by a simple substitution (see Section 4). Hotelling derived the distribution of $T$ when $n_1 = 1$ (in which case $T$ is the generalisation of "Student's" $t$ [4]) and for $m = 2$, [6]. In Section 4, the distribution of $T$ will be derived for arbitrary $m, n_1$ and $n_2$ in the non-central case. More precisely, if $\Omega$ is the matrix of non-centrality parameters, the probability density function of $T$ has the series expansion \begin{equation*}\tag{1}\lbrack\Gamma_m(\frac{1}{2}(n_1 + n_2))/\Gamma(\frac{1}{2}mn_1)\Gamma_m(\frac{1}{2}n_2)\rbrack e^{\mathrm{tr}(-\Omega)}T^{\frac{1}{2}mn_1-1}\end{equation*} $\cdot \sum^\infty_{k = 0}\lbrack(-T)^k/(\frac{1}{2}mn_1)_kk!\rbrack\sum_\kappa (\frac{1}{2}(n_1 + n_2))_\kappa L^\gamma_\kappa(\Omega),$ $\gamma = \frac{1}{2}(n_1 - m - 1),\quad |T| < 1.$ The functions $L^\gamma_\kappa (\Omega)$ are polynomials in the elements of $\Omega$ and are extensions of the classical Laguerre polynomials, to which they reduce when $m = 1$. They will be defined and studied in Section 3. The constants and coefficients occurring in the series are defined in Section 2. If $\Omega = 0$, the density function is \begin{equation*}\tag{2}\lbrack\Gamma_m(\frac{1}{2}(n_1 + n_2))/\Gamma(\frac{1}{2}mn_1)\Gamma_m(\frac{1}{2}n_2)\rbrack T^{\frac{1}{2}mn_1-1}\end{equation*} $\cdot \sum^\infty_{k = 0}\lbrack (-T)^k/(\frac{1}{2}mn_1)_kk!\rbrack \sum_\kappa (\frac{1}{2}n_1)_\kappa (\frac{1}{2}(n_1 + n_2))_\kappa C_\kappa(I),$ where $C_\kappa(I)$ is the zonal polynomial evaluated at the identity matrix (see Section 2). Both series (1) and (2) converge only for $\|T\| < 1$. In the case $m = 1$, the series in (2) reduces to the binomial series for $(1 + T)^{-\frac{1}{2} (n_1+n_2)}$. Unfortunately, this is not very useful since one is usually interested in the upper tail of the distribution. However, it is hoped that the series may be simplified or be the basis for further studies of the distributions. In Section 5, the moments of $T$ are given, again in terms of the generalised Laguerre polynomials in the non-central case. As in the case of the $F$-distribution, only moments of sufficiently low order exist.

In an earlier paper [1] the author presented a generalization of the second degree bipolykays of Hooke [2], defined for arbitrary balanced population structures, and showed the equivalence of these generalized polykays and the $\Sigma$ functions defined by Zyskind [7]. In this paper is presented a more general formalization of generalized symmetric means and polykays of arbitrary degree and some sampling properties of these. Utilizing the fact that the second degree generalized polykays are equivalent to the $\Sigma$'s, which are defined in terms of components of variation, an application to obtaining the variances of estimates components of variation is also presented.

Let $\mathscr{S}(\nu_1, \cdots, \nu_n)$ be the "class of all random variables", $S_n$, which are sums of $n$ independent, non-negative random variables, $X_1, \cdots, X_n$, with $EX_i = \nu_i, i = 1, \cdots, n$. We consider the problem of finding \begin{equation*}\tag{1.1}\inf_{S_n\varepsilon\mathscr{S}(\nu_1,\cdots, \nu_n)} P\{S_n < \lambda\}\end{equation*} where $\lambda$ is a positive constant. For $n = 1$, the infimum is $1 - \nu_1/\lambda$ from the well-known Markov inequality. The solution for $n = 2$ was given in [2]. We derive the solution for $n = 3$. From these results we conjecture what the solution is for arbitrary $n$. To lend support to the conjecture, we examine a sub-class of $\mathscr{S}(\nu_1, \cdots, \nu_n)$, namely those $S_n$'s for which the problem reduces to one of considering the number of successes in independent trials. We show that, within this subclass, the conjectured value does minimize $P\{S_n < \lambda\}$.

In this paper, upcrossings, downcrossings and tangencies to levels and curves are discussed within a general framework. The mean number of crossings of a level (or curve) is calculated for a wide class of processes and it is shown that tangencies have probability zero in these cases. This extends results of Ito [1] and Ylvisaker [7] for stationary normal processes, to nonstationary and non normal cases. In particular the corresponding result given by Leadbetter and Cryer [3] for normal, non stationary processes can be slightly improved to apply under minimal conditions. An application is also given for an important non normal process.

We are concerned with a dynamic system which at times $t = 0, 1, \cdots$ is observed to be in one of a finite number of states. We shall denote the space of possible states by $I$. After each observation the system is "controlled" by making one of a finite number of possible decisions. We shall denote by $K_i$ the set of possible decisions when the system is in state $i, i \varepsilon I$. Let $\{Y_t\}, t = 0, 1, \cdots$, denote the sequence of observed states and $\{\Delta_t\}, t = 0, 1, \cdots$, the sequence of observed decisions. The fundamental assumption regarding $\{Y_t, \Delta_t\}, t = 0, 1, \cdots$, is \begin{equation*}\tag{A}P(Y_{t+1} = j \mid Y_0, \Delta_{0, \cdots,t}Y_t = i, \Delta_t = k) = q_{ij}(k),\quad t = 0, 1, \cdots; j \varepsilon I; k \varepsilon K_i\end{equation*} where the $q_{ij}(k)$'s are non-negative numbers satisfying $\sum_j q_{ij}(k) = 1, k \varepsilon K_i; i \varepsilon I$. A rule for making successive decisions can be summarized in the form of a collection of non-negative functions $D_k(Y_0, \Delta_0, \cdots, \Delta_{t-1}, Y_t),\quad t = 0, 1, \cdots; k \varepsilon K_{Y_t},$ where in every case $\sum_k D_k(\cdot) = 1$. We set $P(\Delta_t = k \mid Y_0, \Delta_0, \cdots, \Delta_{t-1}, Y_t) = D_k(Y_0, D_0, \cdots, \Delta_{t-1}, Y_t)$ for $t = 0, 1, \cdots$. Thus, given $Y_0 = i$ and any rule $R$ for making successive decisions, the sequence $\{Y_t, \Delta_t\}, t = 0, 1, \cdots$, is a stochastic process with its probability measure dependent upon the rule $R$. We refer to such a process as a sequential control process. Let $C$ denote the class of all possible rules; $C'$ denote the class of all rules such that $D_k(Y_0, \Delta_0, \cdots, \Delta_{t-1}, Y_t = i) = D_{ik},\quad t = 0, 1, \cdots; k \varepsilon K_i; i \varepsilon I$. That is, $C'$ is the class of all rules such that the mechanism for making a decision at any time $t$ is dependent only on the state of the system at time $t$. A rule $R \varepsilon C'$ has a stationary Markovian character and, indeed, when $R \varepsilon C'$ is used, the resulting process $\{Y_t\}, t = 0, 1, \cdots$, is a Markov chain with stationary transition probabilities. We let $C''$ denote the subclass of $C'$ where the $D_{ik}$'s are zero or one. Rules in $C'$ allow for randomization; the rules in $C''$ are non-randomized. For a given $R \varepsilon C$ and initial state $Y_0 = i$, let $X_{T,j,k,R}(i) = (T + 1)^{-1}\sum^T_{t=0} P_R(Y_t = j, \Delta_t = k \mid Y_0 = i)$ and let $X_{T,R}(i)$ denote the vector of components $X_{T,j,k,R}(i)$ for all $k \varepsilon K_j$ and $j \varepsilon I$. Denote by $H_R(i)$ the set of limit points of $X_{T,R}(i)$ as $T \rightarrow \infty$. Let $H(i) = \mathbf{\bigcup}_{R\varepsilon C} H_R(i),\quad H'(i) = \mathbf{\bigcup}_{R\varepsilon C'} H_R(i),\quad H''(i) = \mathbf{\bigcup}_{R\varepsilon C''} H_R(i);$ and let $\bar{H}'(i)$ and $\bar{H}''(i)$ denote the convex hulls of $H'(i)$ and $H''(i)$, respectively. In [5] was proved Theorem 1. (a) $\bar H'(i) = \bar H''(i) \supset H(i)$. (b) If the Markov chain corresponding to $R$ is irreducible for every $R \varepsilon C''$, then $\bar H''(i) = H'(i) = H(i) = \mathbf{\bigcup}_{i\varepsilon I} H(i)$. Examples were given in [4] and [5] showing that $H(i)$ can be larger than $H'(i)$. In (b) the irreducibility assumption can be weakened to the condition that for each $R \varepsilon C''$ the corresponding Markov chain has, at most, one ergodic class. Blackwell [1], [2], and Maitra [6] have considered memoryless rules. By a memoryless rule we mean a rule $R$ such that $D_t(Y_0, \Delta_0, \cdots, \Delta_{t-1}, Y_t = i) = D^{(t)}_{ik}\quad t = 0, 1, \cdots, k \varepsilon K_i, i \varepsilon I.$ That is, with a memoryless rule the mechanism for making a decision is a function of the time $t$ and the state $i$ at time $t$. The memoryless rules of Blackwell and Maitra are non-randomized; i.e., $D^{(t)}_{ik} = 0$ or 1. We shall let $C^M$ denote the class of memoryless rules (both randomized and non-randomized). Thus $C \supset C^M \supset C' \supset C''$. If $R \varepsilon C^M - C'$, then $\{Y_t\}, t = 0, 1, \cdots$, is a finite state Markov chain with non-stationary transition probabilities. We remark that it is the memoryless rules (non-randomized) that are considered the rules of interest in the usual finite horizon dynamic programming problems. See Blackwell [2] for interesting remarks along these lines. We are concerned with optimization problems where the criterion to be optimized are functions of the points in $H(i)$. In [5] it was shown that one can construct problems where the optimal rule is in $C - C'$. This can occur, e.g., if the criterion to be optimized is a linear functional over the points in $H(i)$ but where a solution must also satisfy one or more linear constraints in $H(i)$. It is for the purpose of treating optimization problems of this kind that we are interested in the limit points of $X_{T,R}(i)$ for rules belonging to the various important sub-classes of $C$. Let $H^M(i) = \mathbf{\bigcup}_{R\varepsilon C^M} H_R(i)$. The result of this note (which is similar to Theorem 4.1 of [7]) is Theorem 2. $H(i) = H^M(i).$ In fact, for any $R \varepsilon C$ there exists an $R_0 \varepsilon C^M$ such that $X_{tR_0}(i) = X_{tR}(i)$ for all $t$. Proof. Define $R_0$ by $ D_{jk}(t) = P_{R_0}(\Delta_t = k \mid Y_t = j) = P_R(\Delta_t = k \mid Y_t = j, Y_0 = i).$ It is enough to show that for all $t, k \varepsilon K_j$ and $j \varepsilon I$, \begin{equation*} \tag{*} P_{R_0}(Y_t = j, \Delta_t = k \ Y_0 = i) = P_R(Y_t = j, \Delta_t = k \mid Y_0 = i).\end{equation*} The relation $(\ast)$ holds for $t = 0$, since $P_R(Y_0 = j, \Delta_0 = k \mid Y_0 = i) = 0 = P_{R_0}(Y_0 = j, \Delta_0 \mid Y_0 = i)$ if $j \neq i$ and \begin{align*}P_R(Y_0 = i, \Delta_0 = k \mid Y_0 = i) &= P_R(\Delta_0 = k \mid Y_0 = i) \\ &= P_{R_0} (\Delta_0 = k \mid Y_0 = i)\end{align*} by definition. Now assume $(\ast)$ holds for $t = 0, \cdots, T - 1$. Then $P_R(Y_T = j,\Delta_T = k \mid Y_0 = i) = P_R(Y_T = j \mid Y_0 = i)P_R(\Delta_T = k \mid Y_T = j, Y_0 = i)$ but $P_R(\Delta_T = k \mid Y_T = j, Y_0 = i) = D_{jk}(t)$ by definition, and \begin{align*}P_R(Y_T = j \mid Y_0 = i &= \sum_{l\varepsilon I} \sum_{k\varepsilon K_l} P_R(Y_{T-1} = l, \Delta_{T-1} = k \mid Y_0 = i) q_{lj}(k) \\ &= \sum_{l\varepsilon I} \sum_{k\varepsilon K_l} P_{R_0}(Y_{T-1} = l, \Delta_{T-1} = k \mid Y_0 = i) q_{lj}(k) \\ &= P_{R_0}(Y_T = j \mid Y_0 = i)\end{align*} by the introduction hypothesis. Thus \begin{align*}P_R(Y_T = j, \Delta_T = k \mid Y_0 = i) &= P_{R_0}(Y_T = j \mid Y_0 = i)D_{jk}(t) \\ &= P_{R_0}(Y_T = j, \Delta_T = k \mid Y_0 = i).\end{align*} This completes the proof.

Let $X$ and $Y$ be independent random variables having strictly increasing continuous distribution functions $F$ and $G$ respectively. It is shown that for hypotheses of the form $H : F = G$ versus $K_1 : G = F^k, k > 0$ or $H : F = G$ versus $K_2 : G = 1 - (1 - F)^k, k > 0$, the alternative distribution of the MWW $U$-statistic can be generated by a recursive formula analogous to the one used by Mann and Whitney in [2] to generate the null distribution of $U$.

Banerjee [1] has recently given proofs of various properties of real symmetric idempotent matrices. The aim of this note is to give shorter proofs of some of these. Some of our proofs are suggested by the fact that idempotent matrices represent projection operators, and by the known properties of such operators (see e.g. Halmos [2]). The recent paper by Luther [4] overlaps this note in content; but has a different aim.