The purpose of this paper is to present a class of designs suitable for experiments where several responses or characteristics are under study, but all characteristics are not measured on each unit. Usually, while choosing the design, not much consideration has been given to the multiresponse aspect of the experiment, the choice being often made as if only a single response had been under study. Furthermore, while planning multiresponse experiments, we merely take over such designs and then assume that each experimental unit is studied on all variates or characteristics, and carry out the analysis accordingly. However, in a large number of cases, it is either physically impossible, uneconomic, or inadvisable on account of unequal importance or measuring costs of the various characteristics of interest, to study each of them on each experimental unit. Such situations arise in diverse areas of research in the humanities and the natural sciences. An interesting example will be found in [15] and many others in [14]. These communications bring out the important fact that in many situations we need to have experiments where observations on some of the responses are missing not by accident (as, for example, in [7] or [8]), but by design. For the sake of illustration, we include an example here too. Often the process of taking measurements is quite time-consuming. Suppose a biologist has a number of growing organisms of a similar kind, on each of which he could observe (say) $p$ responses provided that the process of taking measurements on any given unit were fast enough to enable him to finish in the limited amount of time, during which the experimental conditions remain unchanged. However, if the process is necessarily slow he may have to content himself with fewer (than $p$) responses on each unit. The above indicates that every design has two aspects: one relative to the responses and experimental units, and the other relative to treatments and blocks. The first decides for each unit the responses to be studied therein, while the latter tells us for each block the treatments (and units) allocated to it. However, the first aspect may also influence the second in the sense that relative to each response, we may have a different system of blocks. Designs which possess this last property are called multiresponse designs with $p$ block systems. These have been studied in [11] and will not be considered here. In [11], we have also studied another class called hierarchical designs, which are suitable for those situations in which the various responses could be graded in a descending order of importance, say $(V_1, V_2, \cdots)$, such that the response $V_i$ is supposed to be more important than $V_j$, if $i < j$. The designs are defined essentially by requiring that if $V_i$ is more important than $V_j$, then $V_i$ should also be observed on each experimental unit on which $V_j$ is observed. However, in many cases such a grading may not exist, or it may be otherwise inadvisable to impose such a hierarchical structure on the experiment. More general classes of designs are then called for. In this paper we introduce a class of designs (which may be called `regular incomplete multiresponse designs') that are response-wise incomplete, i.e., in which there are (at least some) experimental units on which all responses are not measured. However, in addition, they may or may not be treatment-wise incomplete (in the sense of an ordinary incomplete block design). As will be evident from the definition in the next section, these designs will be useful in a large variety of multiresponse experiments. The approach to the analysis of response-wise incomplete designs, by Trawinski (unpublished dissertation, [14]) and Trawinski and Bargmann [15], is through the use of maximum likelihood estimates and the likelihood ratio tests. However, as mentioned by the author in [13], apart from large sample approximations, etc., (inherent in the likelihood ratio tests), the formulae for the estimates of the parameters and for the test statistic that one obtains this way are somewhat cumbersome even for an electronic computer. In this paper therefore, a different approach which works for regular designs, is adopted. The attempt is to transform the data back into the framework of linear estimation and multivariate analysis of variance. Once this is done, the usual techniques of analysis become available. It may be stressed that the purpose of the regular designs is not merely to permit the estimation of treatment effects by linear analysis methods (which in itself is important), but also to make available valid (free of nuisance parameters) and exact test regions for testing linear hypotheses on the treatments. As usual, we require normality assumptions for the latter but not the former. It will also be noted that the approach in this paper possesses in a sense a symmetry with respect to the responses, unlike, for example, the one used in the analysis of hierarchical designs in [11] where the responses are arranged in an order of importance, say $V_1, V_2, \cdots, V_p$, and then the analysis for $V_{i+1} (i = 0, \cdots, p - 1)$ is performed conditional to the data observed for $V_1, V_2, \cdots, V_i$. Finally, we remark that though the derivations of the condition for regular designs may seem a little complex, their actual use is simple relative particularly to the other existing techniques.

Mehler's identity is used to obtain a bound for the integral of the absolute difference between the bivariate gaussian density function and its corresponding marginal densities. In a sense, this integral measures the contribution of the dependency between the gaussian random variables to an expectation. It is shown that the integral is dominated by $|\rho|/ (1 - |\rho|)$, where $\rho$ is the correlation coefficient between the random variables. Using the Hotelling canonical decomposition of a variance-covariance matrix, the result is extended to the case of dependent gaussian vectors with the bound now given in terms of the canonical correlations, i.e., the roots of a characteristic equation related to the variance-covariance matrix of the vectors. As an application of the results, a bound is obtained for the variance of the function $F(X_1, \cdots, X_n) = n^{-1} \sum^n_{i=1} f(X_i)$. The $\{X_i\}$ denote a sequence of dependent, non-stationary, gaussian random variables (or vectors), and $f(x)$ is any bounded measurable function. For the stationary case, the rate of convergence of the variance is easily expressed in terms of the summability properties of the correlation coefficients. The paper concludes with some comments on extending the results to the class of $\varphi^2$-bounded bivariate density functions.

Let $\{X(t), t = \cdots -1, 0, 1, \cdots\}$ be a $P$ dimensional zero mean stationary Gaussian time series, $X(t) = \begin{pmatrix}X_1(t)\\X_2(t)\\\vdots\\X_P(t)\end{pmatrix}$ we let $R(\tau) = EX(t)X' (t + \tau)$, where $R(\tau) = \{R_{ij}(\tau), i,j = 1, 2, \cdots P\}$, and $F(\omega) = (2\pi)^{-1} \sum^\infty_{\tau=-\infty}e^{-i\omega\tau}R(\tau)$. It is assumed that $\sum^P_{i,j=1} \sum^\infty_{\tau=-\infty} |\tau| |R_{ij}(\tau)| < \infty$, and hence $F(\omega)$ exists and the elements possess bounded derivatives. It is further assumed that $F(\omega)$ is strictly positive definite, all $\omega$. Knowledge of $F(\omega)$ serves to specify the process. $F(\omega)$, and $S$, the covariance matrix of $x = \begin{pmatrix}x_1 \\ x_2\ \\ vdots\\x_P\end{pmatrix}$, a Normal $(0, S)$ random vector are known to enjoy many analogous properties. (See [7].) To cite two examples, the hypothesis that $X_i(s)$ is independent of $X_j(t)$ for $i \neq j = 1, 2, \cdots P$, any $s, t$, is equivalent to the hypothesis that $F(\omega)$ is diagonal, all $\omega$, while the hypothesis that $x_i$ is independent of $x_j$, for $i \neq j = 1, 2, \cdots P$ is equivalent to the hypothesis that $S$ is diagonal. The conditional expectation of $x_1$, given $x_2, \cdots x_P$ is \begin{equation*}E(x_1\mid x_2, \cdots x_P) = S_{12}S^{-1}_{22}\begin{pmatrix}x_2 \\ \vdots \\ x_P\end{pmatrix}, S = \bigg(\begin{array}{c|c} S_{11} & S_{12} \\ \hline S_{21} & S_{22}\end{array} \bigg)\end{equation*}. The corresponding regression problem for stationary Gaussian time series goes as follows. If \begin{equation*}E\{X_1(t)\mid X_2(s), \cdots X_P(s), s = \cdots -1, 0, 1, \cdots\} = \sum^P_{j=2} \sum^\infty_{s=-\infty} b_j(t - s)X_j(s)\end{equation*} then $B(\omega)$, defined by $B(\omega) = (B_2(\omega), \cdots B_P(\omega)), B_j(\omega) = \sum^\infty_{s=-\infty} b_j(s)e^{i\omega s}$ satisfies \begin{equation*}B(\omega) = F_{12}(\omega)F_{22}^{-1}(\omega), \quad F(\omega) = \bigg(\begin{array}{c|c}f_{11}(\omega) & F_{12}(\omega) \\ \hline F_{21}(\omega) & F_{22}(\omega)\end{array} \bigg).\end{equation*} It is interesting to ask how well these and similar analogies carry over to sampling theory and hypothesis testing. Goodman [3] gave a heuristic argument to support the conclusion that $\hat{F}_X(\omega_k)$, a suitably formed estimate of the spectral density matrix $F(\omega_k)$ has the complex Wishart distribution. The question is met here by the following results. Firstly if $\hat{F}_X(\omega_l), l = 1, 2, \cdots M$ are estimates of the spectral density matrix, each consisting of averages of $(2n + 1)$ periodograms based on a record of length $T$, with the $\omega_l$ equally spaced and $(2n + 1)M \leqq \frac{1}{2} T$, then it is possible to construct, on the same sample space as $X(t), M$ independent complex Wishart matrices $\hat{F}{\bar{\bar{X}}}(\omega_l), l = 1, 2, \cdots M$ such that $\{\hat{F}_X(\omega_l), l = 1, 2, \cdots M\}$ converge simultaneously in mean square to $\{\hat{F}_{\bar{\bar{X}}}(\omega_l), l = 1, 2,\cdots M\}$, as $n, M$ get large. Secondly, it is legitimate to use the natural analogies from multivariate analysis to test hypotheses about time series. One example is presented, as follows. The likelihood ratio test statistic for testing $S$ diagonal is $|\hat{S}|/\mathbf{\prod}^P_{i=1} \hat{s}_{ii}$ where $\hat{S} = \{\hat{s}_{ij}$ is the sample covariance matrix. The analogous statistic $\psi$ for testing $X_i(s), X_j(t)$ independent, $i,j 1 = 2, \cdots P$ from a record of length $T$ is $\psi = \prod^M_{l=1} \lbrack|\hat{F}_X(\omega_l)|/\prod^P_{i=1} \hat{f}_{ii}(\omega_l)\rbrack$ where $\hat{F}_X(\omega_l) = \{\hat{f}_{ij}(\omega_l)\}$ are the sample spectral density matrices as above. Letting ${\bar{dbar{\psi}}} = \prod^M_{l=1} \lbrack|\hat{F}_{\bar{\bar{x}}}(\omega_l)|/\prod^P_{i=1} \hat{h}_{ii}(\omega_l)\rbrack$ where $\hat{F}_{\bar{\bar{x}}}(\omega_l) = \{\hat{h}_{ij}(\omega_l)\}$ are the independent complex Wishart matrices referred to above, we show $EC_{n,M} |\log \psi - \log {\bar{\bar{\psi}}} \rightarrow 0$ for large $n, M$, where $C_{n,M}$ are chosen to make the result non-trival. The method of proof applies to any statistic which is a product over $l$ of sufficiently smooth functions of the entries of $\hat{F}_X(\omega_l)$. Applications to estimation and testing in the regression problem will appear elsewhere [8]. The distribution theory of functions of complex Wishart matrices has been well investigated by a number of authors [3] [5] [6], and hence can be easily applied here to statistics like ${\bar{\bar{\psi}}}$. The results above are shown for $P = 2$, it is clear that the proofs extended to any (fixed) finite $P$. The proofs proceed as follows, via a theorem which has somewhat more general application. For each $T$, let $X$ be the $2 \times T$ random matrix $X = \binom{X_1}{X_2} = \begin{pmatrix}X_1(1), \cdots, X_1(T)\\X_2(1), \cdots, X_2(T)\end{pmatrix}$ and let the $2T \times 2T$ covariance matrix $\Sigma$ be given by $\Sigma = \begin{pmatrix}\sum_{11} \sum_{12} \\ \sum_{21} \sum_{22}\end{pmatrix}$ where $\Sigma_{ij} = EX_i'X_j. \{\hat{F}_X(\omega_l)\}$, the sample spectral density matrices described above based on a record of length $T$, are each of the form $\hat{F}_X(\omega_l) = T^{-1}XQX'$ where $Q$ is a $T \times T$ circulant matrix with largest eigenvalue $ = T(2n + 1)^{-1} \leqq \frac{1}{2}M < <T$. We define circulant matrices $\bar{\Sigma}_{ij}$ which approximate $\Sigma_{ij}$, and a random matrix $\bar{X}$ on the sample space of $X$, $\bar{X} = \binom{\bar{X}_1}{\bar{X}_2} = \begin{pmatrix}\bar{X}_1(1), \cdots, \bar{X}_1(T)\\\bar{X}_2(1), \cdots, \bar{X}_2(T)\end{pmatrix}$ with $E\bar{X}_i'\bar{X}_j = \bar\Sigma_{ij}$. The $2T$ eigenvalues of the block circulant matrix $\bar\Sigma = \begin{pmatrix}\bar\Sigma_{11} \bar\Sigma_{12} \\ \bar\Sigma_{21} \bar\Sigma_{22}\end{pmatrix}$ will be the $2T$ eigenvalues of the $T$ matrices $\{F(2\pi j/T),j = 1, 2, \cdots T\}$. The distribution of random matrices of the form $T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is any circulant matrix are relatively simple to investigate due to the fact that all circulant matrices commute, and their eigenvalues may be exhibited as simple functions of the elements. Circulant quadratic forms in random vectors with circulant covariance matrices are well known in the literature, (See [1] and references cited there). Let $\hat{F}_{X,Q} = T^{-1}XQX'$ and $\hat{F}_{\bar{X},Q} = T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is now any $T \times T$ (real or complex) quadratic form with largest absolute eigenvalue $\leqq q$. The main Theorem allows the replacement of $X$ by $\bar{X}$ in the analysis, and is, that under the assumptions on $F(\omega)$ and $R(\tau)$, for any $T$, \begin{equation*}\tag{1.1} E \operatorname{tr} (\hat{F}_{X,Q} - \hat{F}_{\bar{X},Q})(\hat{F}_{X,Q} - \hat{F}_{\bar{X}, Q})^{\ast'} \leqq cq^2/T^2\end{equation*} where $c$ is a constant depending only on $F(\omega)$ and $R(\tau)$. A lemma, essentially allowing the replacement of $F(\omega)$ by a suitably chosen step-function, together with the application of (1.1) gives the results concerning the $\{\hat{F}_X(\omega_l)\}$ an $\lambda$. Since $\hat{R}(\tau)$, the sample (circularized) autocorrelation function is also of the form $T^{-1}XQX'$ with $Q$ circulant we obtain an easy corollary on the distribution of $\{\hat{R}(\tau)\}$.

For the random variables $X_{ij}(i = 1, \cdots, N; j = 1, \cdots, r)$ consider the linear model \begin{equation*}\tag{1.1} X_{ij} = \mu + \beta_i + \tau_j + Y_{ij} (\sum \beta_i = 0, \sum\tau_j = 0),\end{equation*} where the $\tau$'s are treatment effects, the $\beta$'s are nuisance parameters (block effects), and the $Y_{ij}$'s are error components. Nonparametric procedures for estimating and testing contrasts in the $\tau$'s, based on the Wilcoxon signed rank statistics, are due to Lehmann (1964), Hollander (1967) and Doksum (1967), among others. These rest on the assumption that the $Y_{ij}$'s are independent with a common continuous distribution. Since these procedures are actually based on the paired differences $X^\ast_{ijk}$, defined by (2.1), they are unaffected by the addition of a random variable $V_i$ to $\beta_i$ (or to $Y_{ij}$) for $i = 1, \cdots, N$. The object of the present investigation is to show that these procedures are valid even if $Y_{i1}, \cdots, Y_{ir}$ are interchangeable random variables, for each $i( = 1, \cdots, N)$. It may be noted that if in (1.1) the superimposed random variable $V_i$ is absorbed in $Y_{ij}$, then of course $Y_{i1}, \cdots, Y_{ir}$ are interchangeable, but the interchangeability of $Y_{i1},\cdots, Y_{ir}$ does not necessarily imply that $Y_{ij} = W_{ij} + V_i$, where $W_{ij}$'s are independent and identically distributed random variables (iidrv). In fact, in `mixed model' experiments, interchangeability of $Y_{i1}, \cdots, Y_{ir}$ (of quite arbitrary nature) may arise when there is no block versus treatment interaction [cf. Koch and Sen (1968) for details]. It is also shown that the procedures mentioned above are robust against possible heterogeneity of the distributions of the error vectors $\mathbf{Y}_i = (Y_{i1}, \cdots, Y_{ir}), i = 1, \cdots, N$. This situation may arise when the block effects are not additive or the errors are heteroscedastic. Thus, in this paper the independence of the errors is replaced by within block symmetric dependence, while the additivity of the block effects and homoscedasticity of the errors are relaxed.

In the definition of a two-person zero-sum game given by Von Neumann and Morgenstern it is assumed that both players know the rules of the game (e.g., the game tree, the information sets as well as the distributions of the ensuing payoffs for given strategy choices, etc.). We use the term pseudo-game to denote the case where at least one player does not have complete information. In this paper we restrict our attention to those pseudo-games in which player I, say, is only aware of his set of pure strategy choices (assumed to contain $m$ elements: $2 \leqq m < \infty$) and not of player II's strategy choices (assumed to have uniformly bounded second moments). Player II is assumed to have complete information. More precisely, we shall study pseudo-games $G$ that have the format given below: Let $A = \{a_1, \cdots, a_m\}$ denote the pure strategy choices of player I. Denote by $A^\ast$ the set of probability distributions $p$ over $A$ (player I's mixed strategy choices). We sometimes write $p$ in the form $(p(1), \cdots, p(m)), \sum^m_{j=1}p(j) = 1$, and $p(j) \geqq 0$, with the interpretation that when player I uses $p$ he will play $a_j$ with probability $p(j)$. Any element of $A^\ast$ that assigns mass 1 to some $a \varepsilon A$ will be simply denoted by $a$. Let $B$ denote the set (not necessarily finite) of pure strategies for player II. Let $\mathscr{B}$ be a fixed $\sigma$-field of subsets of $B$ and denote by $B^\ast$ the set of all probability distributions $q$ over $\mathscr{B}$ (player II's mixed strategies). We assume that $\mathscr{B}$ contains all single point sets of $B$, so that $B^\ast$ contains all finite probability distributions over $B$. We postulate that we are given for each pair $(a, b)$ in the product space $A \times B$ a distribution $P_{(a, b)}$ on the real line which represents the distribution of the loss incurred by player I (or gain by player II) if $a \varepsilon A$ is the strategy choice of I and $b \varepsilon B$ is the strategy choice of II. Contrary to the usual practice, the payoff for given pure strategy choices is thus allowed to be random. We do this in order that our main results may be proved in greater generality. An example of a pseudo-game with random payoffs is given in Section 2. The distributions $P_{(a,b)}$ are assumed to have uniformly bounded second moments. For each $a \varepsilon A$, and fixed Borel set $C, P_{(a,\cdot)}(C)$ is assumed to be $\mathscr{B}$-measurable. For each pair $(a, b) \varepsilon A \mathbf{\times} B$, let $X_{(a,b)}$ be a random variable having $P_{(a,b)}$ as its distribution. Suppose that players I and II are using strategies $p$ and $q$, respectively. They can determine the payoff of the pseudo-game by first selecting an $a \varepsilon A$ and a $b \varepsilon B$ according to the distributions $p$ and $q$, respectively, and then treating an observed value of $X_{(a,b)}$ as the payoff. For every pair of strategies $(p, q)$ that players I and II may use we define the expected value of the payoff $R(p, q)$ by means of the equation: \begin{equation*}\tag{1.1} R(p, q) = \sum^m_{j=1} p(j) \int \lbrack\int x dP_{(a_j,b)}(x)\rbrack dq(b).\end{equation*} We are assuming that player I is only aware of the set $A$, while player II has complete information. However, by assuming instead that player I is also aware of the set $B$ as well as the distributions $P_{(a,b)}, (a, b) \varepsilon A \times B$, we can associate with every such pseudo-game $G$ a game with complete information $G'$. Such concepts as "value" and "minimax strategy" do not carry over to pseudo-games. However by the minimax theorem, since $A$ is assumed finite, every such game $G'$ will have a value $v_G$ and player I will have a minimax strategy $p'$: \begin{equation*}\tag{1.2)} v_G = \sup_{q\varepsilon B^\ast} R(p', q) = \inf_{p\varepsilon A^\ast} \sup_{q\varepsilon B^\ast} R(p, q) = \sup_{q\varepsilon B^\ast} \inf_{p\varepsilon A^\ast} R(p, q).\end{equation*} Suppose now that players I and II are playing a sequence of identical pseudo-games of the type we have been describing; i.e., they play one game, observe their losses and play the same game again (with possibly different strategy choices), continuing in this manner ad-infinitum. We shall refer to the individual games that make up the sequence as the subgames of the sequence. When playing such a sequence of pseudo-games a strategy for player I would be a rule $P$ that would tell him for every $j$, as a function of his past plays (mixed strategy choices) and losses what mixed strategy to play during the $j$th subgame; a strategy for player II would be a rule $Q$ that would tell him for every $j$, as a function of his own, and his opponent's past plays and losses, what mixed strategy $q$ to play during the $j$th subgame of the sequence. We are thus allowing player II to know what plays player I has made, but we are not granting I the same favor. Among the rules $P$ available to player I we define a special class of rules to be called rules constant on intervals. If $x$ is any real number let $\lbrack x\rbrack$ denote the largest integer that is less than or equal to $x$. For every $\alpha > 1$, let $\Pi(\alpha) = (I_1(\alpha), I_2(\alpha), \cdots, I_n(\alpha), \cdots)$ denote the partition on the set I of positive integers defined by the equations: \begin{equation*}\tag{1.3} I_n(\alpha) = \{(\sum^{n-1}_{k=1} \lbrack k^\alpha\rbrack) + 1, (\sum^{n-1}_{k=1}\lbrack k^\alpha\rbrack) + 2, \cdots, \sum^n_{k=1} \lbrack k^\alpha\rbrack\}; n = 1, 2, 3,\cdots.\end{equation*} For example, $I_1(2) = \{1\}, I_2(2) = \{2, 3, 4, 5\}, I_3(2) = \{6, 7, \cdots, 14\}$; etc. We shall refer to $I_n(\alpha)$ as the $n$th interval of the partition $\Pi (\alpha)$. Note that the cardinality of $I_n(\alpha)$ is $\lbrack n^\alpha\rbrack$. Let us suppose that player I is using some rule $P$ that assigns, with probability 1, the same mixed strategy to the $i$th subgame as it does to the $j$th subgame whenever $i$ and $j$ belong to the same interval $I_n(\alpha), n = 1, 2, 3, \cdots$. In this case we say that $P$ is constant on intervals. Thus if we say that player I is to play a certain strategy $p$ during the $n$th interval of a partition $\Pi(\alpha)$, we mean that he is to play $p$ during every subgame whose index belongs to $I_n(\alpha)$. The particular strategy that player I uses in the $n$th interval (a random variable depending on plays and losses occurring prior to the $n$th interval) will be denoted by $p_n$. For $j = 1, 2, 3, \cdots, N, \cdots$ let $X_j$ represent the loss incurred by player I during the $j$th subgame. Note that the sequence $\{X_n\}$ is a discrete stochastic process whose index set is the set I of positive integers and whose law of evolution is determined by the distributions $P_{(a,b)}$ and by the rules $P$ and $Q$ that the players use. The first objective of this paper is to prove: THEOREM. Suppose players I and II are playing a sequence of identical pseudo-games $G$ satisfying (i) and (ii): (i) Player I has $m \geqq 2$ pure strategy choices. (ii) The distributions $P_{(a,b)}$ have uniformly bounded second moments and for each $a \varepsilon A$ and every Borel set $C, P_{(a,\cdot)}(C)$ is $\mathscr{B}$-measurable. Then there exists a class of rules $\{P\}_m$ for player I such that for all rules $Q$ that player II may use we have: $P \varepsilon \{P\}_m \Rightarrow \operatorname{Pr} (\lim \sup_{N\rightarrow\infty} N^{-1} \sum^N_{j=1} X_j \leqq v_G\mid P, Q) = 1.$ We will show, that is, that the player with incomplete information can do as well asymptotically as he could if he had complete information. The members of $\{P\}_m$ will all be constant on intervals. Our second objective will be to seek a strong convergence rate for $N^{-1} \sum^N_{j=1} X_j$. In the course of achieving this goal we will show that a good partition is obtained by setting $\alpha$ equal to $(m + 2)/m$.

It is shown that the mean of a normal distribution with unknown variance $\sigma^2$ may be estimated to lie within an interval of given fixed width at a prescribed confidence level using a procedure which overcomes ignorance about $\sigma^2$ with no more than a finite number of observations. That is, the expected sample size exceeds the (fixed) sample size one would use if $\sigma^2$ were known by a finite amount, the difference depending on the confidence level $\alpha$ but not depending on the values of the mean $\mu$, the variance $\sigma^2$ and the interval width $2d$. A number of unpublished results on the moments of the sample size are presented. Some do not depend on an assumption of normality.

The problem of sequential selection of experiments, with fixed and optional stopping, is considered. Conditions are given which allow selection, stopping and terminal action rules to be based on a sequence $\{T_j\}$ of statistics, where $T_j$ is a function of past observations $\mathbf{X}^j = (X_1, \cdots, X_j)$ and experiment selections $\mathbf{E}^j = (E_1, \cdots, E_j)$. Randomized stopping, selection, and terminal action rules are allowed, and all probability distributions are defined by densities relative to $\sigma$-finite measures over Euclidean spaces. Here we give a heuristic description of the principal results for the case of optional stopping. At each time $j$ the random variable $X_j$ is observed and a decision is made to stop or continue. If the procedure is stopped, a terminal action $A$ is taken. If it is continued, an experiment $E_{j+1}$, to be performed at time $j + 1$, is chosen. At time $j$, all decisions are based on $\mathbf{X}^j,\mathbf{E}^j$, the past observations and experiment selections. Upon stopping, and taking action $A$, a loss $L(\theta, A)$, where $\theta$ is the unknown state of nature, is incurred. The sampling cost of stopping at $j$ is $C_j(\theta, \mathbf{X}^j, \mathbf{E}^j)$. Let the random variable $N$ denote the random stopping time. A selection rule $\gamma = (\gamma_0, \gamma_1, \cdots)$ is defined by the sequence of conditional densities $\gamma_j(e_{j+1}\mid\mathbf{x}^j, \mathbf{e}^j)$, a stopping rule $(\mathbb{\Phi} = (\phi_0, \phi_1, \cdots)$ by the probabilities $\phi_j(\mathbf{x}^j,\mathbf{e}^j) = P\{N = j\mid N \geqq j, \mathbf{x}^j,\mathbf{e}^j\}$, and a terminal action rule $\delta = (\delta_0, \delta_1, \cdots)$ by the conditional densities $\delta_j(a\mid\mathbf{x}^j,\mathbf{e}^j)$. Definition of the population densities $f_\theta(x_{j+1}\mid\mathbf{x}^j, \mathbf{e}^{j+1})$ for $j = 0, 1, 2, \cdots$ completely fixes the probability structure. Define $\{T_j\}$ to be parameter sufficient (PARS) if, for $j = 0, 1, 2, \cdots, \operatorname{Dist}_{\theta,\gamma}(\mathbf{X}^j, \mathbf{E}^j\mid T_j)$ is independent of $\theta$ for all $\gamma$ and policy sufficient (POLS) if, for $j = 0, 1, 2, \cdots, \operatorname{Dist}_{\theta,\Phi,\gamma} (T_{j+1}\mid T_j, E_{j+1}, N \geqq j + 1)$ is independent of $\mathbf{\phi}, \mathbf{\gamma}$ for all $\theta$. THEOREM. If $\{T_j\}$ is PARS; then the class of policies $\{\mathbf{\phi}, \mathbf{\gamma}, \mathbf{\delta}^0\}$, where $\delta^0$ is based on $\{T_j\}$, is essentially complete. THEOREM. If $\{T_j\}$ is PARS and POLS, and the sampling cost is of the form $C_j(\theta, T_j)$, then the class of policies $\{\mathbf{\Phi}^0, \mathbf{\gamma}^0, \mathbf{\delta}^0\}$, where $\mathbf{\phi}^0, \mathbf{\gamma}^0, \mathbf{\delta}^0$ are based on $\{T_j\}$, is essentially complete. Conditions are given to aid in the verification of PARS and POLS. The theorems are applied to examples, including versions of the two armed bandit problem.

This note provides some new second order rotatable designs. The method of construction used is an extension of one introduced by Bose and Draper (1959). Further extensions of the method are briefly suggested.

This paper is the result of an attempt to clarify and improve some results in the theory of statistical information. The term information is used to denote different things in different contexts. First of all, there is Shannon's information, $-\sum p_i \log p_i$, defined for probability distributions on a finite sample space; this measures, in an esthetically satisfactory way, the entropy or amount of uncertainty in a distribution. Then there is Wiener's information, $\int f(x) \log f(x) dx$, defined for an absolutely continuous distribution on the line (or in $n$-space); it was introduced by Wiener, with an acknowledgment to von Neumann, as a "reasonable measure" of the amount of information, having the property of being "the negative of the quantity usually defined as entropy in similar situations" ([10], p. 76). Finally, there is "information of one probability distribution $P$ with respect to another $Q$," commonly known as Kullback-Leibler information. On a finite sample space, this has the form $\sum p_i \log (p_i/q_i) = - \sum p_i \log q_i - (- \sum p_i \log p_i)$, and thus has some relationship to entropy; note that the second term, which is the entropy of $\{p_i\}$, is the minimum of the first expression over all distributions $\{q_i\}$. An interesting idea due to Gelfand, Kolmogorov and Yaglom [3] establishes a connection between the Kullback-Leibler information for a finite probability space and that for any space: If $P, Q$ are probability measures on a measurable space $(\Omega, \mathscr{F}), P \ll Q$, and $\{A_i, i = 1, \cdots, n\}$ is any finite measurable partition of $\Omega$, then the supremum of $\sum_i \log \lbrack P(A_i)/Q(A_i)\rbrack P(A_i)$ over all finite measurable partitions is $\int_\Omega \log (dP/dQ)dP$. The only published proof of this result seems to be that due to Kallianpur [5], which uses martingale theory. In Section 1, we shall obtain a rather simple direct proof of this result (Theorem 1.1) and extend it to the case where $Q$ is any $\sigma$-finite measure (Theorem 1.2). Wiener's information is then seen to be the supremum of $\sum \log \lbrack P(A_i)/Q(A_i)\rbrack P(A_i)$ over countable partitions, with $Q =$ Lebesgue measure. Section 2 will be concerned with Kullback-Leibler information. We shall define conditional information relative to a sub-field, establish a relation between this conditional information and sufficiency of the sub-field (Theorem 2.2), and also show that this conditional information equals the difference between information contained in the field and that in the sub-field (Theorem 2.3). These are extensions of results obtained by Kullback and Leibler in a somewhat limited context.

A decision maker is confronted with $k$ populations, $\pi_1, \cdots, \pi_k$, (say, $k$ lots of items available for purchase) and a control population $\pi_0$ and must, on the basis of random samples of common size $n$ from $\pi_0,\cdots, \pi_k$, select those which are at least as good as $\pi_0$. We suppose that items are judged on the basis of a continuously distributed attribute $X$ and that a known fraction $\alpha (0 < \alpha < 1)$ of the items in the control population are deficient (their $X$-values are too small). A population is considered to be better than the control if it has a smaller proportion of deficient items; that is, letting $F_j, j = 0,\cdots, k$, denote the distribution function (df) of $X$ for population $\pi_j$ and $x_\alpha(F_j)$ its $\alpha$th quantile, $\pi_j$ is better than $\pi_0$ if $x_\alpha(F_j) \geqq x_\alpha(F_0)$. We also consider the possibility that $F_0$ is known in which case $\pi_0$ is called a standard and is not sampled. In Section 2 we propose a nonparametric procedure $R$ based on order statistics which guarantees a minimal preassigned probability $P^\ast$ that, when each $F_j$ is stochastically ordered with respect to $F_0$, all populations better than the control will be selected; such a selection will be called a correct selection (CS). The corresponding problem of selecting a subset containing the best population (without any control) was treated in [11]. Since the trivial procedure $R_0$ of including all $k$ populations in the selected subset also guarantees the probability requirement it is necessary to investigate the expected number of misclassifications; this is done exactly in Section 3 and asymptotically in Section 5. Exact results for known standard $F_0$ are given in Section 4. Some other aspects of the problem are briefly discussed in Section 8. As a secondary problem we suppose that for some preassigned fraction $\delta^\ast$ the decision maker considers a population $\pi_i$ to be $\delta^\ast$-inferior to $\pi_0$ if more than $100(\alpha + \delta^\ast)$ percent of the items in $\pi_i$ are as bad as at least one of the worst $100(\alpha - \delta^\ast)$ percent of the items in $\pi_0$; i.e., $\pi_i$ is $\delta^\ast$-inferior if $x_{\alpha-\delta}^\ast(F_0) \geqq X_{\alpha+\delta\ast}(F_i)$. In Section 5 we give asymptotic expressions for the smallest sample size needed to guarantee that the expected proportion of $\delta^\ast$-inferior populations selected by $R$ will be less than a preassigned number $\beta^\ast$. An equally reasonable definition of $\pi_i$ to be $\delta^\ast$-inferior is that more than $100(\alpha + 2\delta^\ast)$ percent of the items in $\pi_i$ are deficient. Our results with $\alpha$ replaced by $\alpha' = \alpha + \delta^\ast$ also apply to this problem. We show in Section 6 that for small values of $\delta^\ast$ a competing non-parametric procedure $S$ based on rank sums and a competing asymptotically non-parametric procedure $M$ based on sample means both require sample sizes proportional to the square of that required by $R$ to achieve the same degree of rejection of $\delta^\ast$-inferiors. For moderate $\delta^\ast$-values it is shown that $S$ requires a sample size which has the same order of magnitude as that required by $R$. In Section 7 we study a related minimax procedure. We append tables for $\alpha = \frac{1}{2}$ of (1) the integer constant $c$ needed to make procedure $R$ explicit, (2) some required values to make the minimax procedure explicit and (3) efficiency comparisons of $S$ with respect to $R$. A Basic Inequality. Let $\mathbf{X} = \{X_{ji}, 1 \leqq j \leqq n, 0 \leqq i \leqq k\}$ denote the combined sample, thus for each $i, X_{1i}, \cdots, X_{ni}$ are independent random variables having the $df F_i(x)$. We regard $\omega = (F_0, F_1,\cdots, F_k)$ as the unknown "parameter" and, for an arbitrary function $\psi$, use the symbol $E_\omega\psi(\mathbf{X})$ to denote the expected value of $\psi(\mathbf{X})$ computed under the assumption that $\omega$ is the true parameter value. The following lemma is used extensively in this paper; we state it without proof since it follows easily from Lemma 2.1 of [1]. LEMMA 1.1 Let $\psi(\mathbf{x})$ be non-increasing in each $x_{j0}, j = 1,\cdots, n$, and non-decreasing in each $x_{ji}, 1 \leqq j \leqq n, 1 \leqq i \leqq k$, and let $\omega = (F_0, F_1,\cdots, F_k)$ and $\omega' = (F'_0, F'_1,\cdots, F_k)$ satisfy $F_0(x) \leqq F'_0(x)$ and $F_i(x) \geqq F'_i(x)$ for $i = 1,\cdots, k$ and all $x$, then $E_\omega\psi(\mathbf{X}) \leqq E_{\omega'}\psi(\mathbf{X})$.

In the "two-armed-bandit with finite memory" problem, each rule which has been proposed (see [2], [3], and [4]) can be improved by using a corresponding randomized rule. The performance of various randomized rules is computed.

Let $X_{(1)} < \cdots < X_{(n)}$ and $Y_{(1)} < \cdots < Y_{(n)}$ be the order statistics of two independent random samples from the absolutely continuous distribution functions $F(x)$ and $G(y)$, respectively. Let $T_n$ be the proportion of pairs, $(X_{(i)}, Y_{(i)})$, for which $X_{(i)} \geqq Y_{(i)}$. Tests of the equality of $F$ and $G$ based on $T_n$ are among the oldest nonparametric procedures in the literature, going back at least to Galton's analysis of Darwin's data [3]. Hodges [5] showed the null distribution of $nT_n$ to be uniform over $0, 1,\cdots, n$. Bickel and Hodges [1] treated the asymptotic distribution of the Lehmann estimate based on the one-sample version of $T_n$. In this note we use very elementary methods to derive expressions for the distribution and moments of $T_n$ from which conditions for the consistency of tests based on $T_n$ follow immediately. More generally we can show that (unnormalized) $T_n$ always has an asymptotic distribution for any pair $(F, G)$. This distribution is degenerate at zero if $Y$ happens to be stochastically larger than $X$. We give informative expressions for the first two moments of this asymptotic distribution. Our technique is to express the distribution of $T_n$ in terms of integrals of certain multinomial probabilities.

The rapid development of non-parametric rank tests was generated, in part, by the result of Hodges and Lehmann [6] which stated that the asymptotic relative efficiency (ARE) of the Wilcoxon test to the classical two-sample $t$-test was always $\geqq.86$. They also conjectured that the ARE of the normal scores test to the $t$-test was always greater than or equal to one. In 1958, Chernoff and Savage [3] proved the validity of this conjecture using variational methods. In this paper we give a simple proof of their result. Recently Doksum [4] has shown that the Savage test [11] maximizes the minimum asymptotic power for testing for scale change over the family of distributions with increasing failure rate averages (IFRA) [2]. The technique of our proof enables us to obtain a lower bound for the asymptotic power of Savage's test for scale change of any positive random variable possessing a finite second moment. When the positive random variables are restricted to be IFRA, Doksum's [4] results follow.

Consider a production process which exists in one of two states, in-control and out-of-control. Production begins in-control and while there a chance event occurs after each item is produced so that the probability of remaining in-control is $(1 - \pi)$ and the probability of a transition out-of-control is $\pi$. Once out-of-control the process remains there until trouble is removed. Associated with each item produced is a measurable characteristic or quality which is a Gaussian random variable whose mean depends on the underlying process state. Let the mean in-control be $\mu_0$, the mean out-of-control be $\mu_1$ and the common variance be $\sigma^2$. It costs $K$ units to repair the process and the operating cost rate in-control is $c_0$ and out-of-control, $c_1$. The true process state is unknown and a rule is desired which specifies when to repair, based on the quality history, so as to minimize some function of the cost components. This model of a production process was first studied by Girshick and Rubin (1952). They show that when the criterion is long run time average total cost to repair plus operation, the optimal rule is of the form: "Stop and repair after production of the $k$th item if and only if $Z(k) \geqq \zeta$" for some critical value $\zeta$, where $Z(k)$ is a monotonic function of the posterior probability, given the observed history, that the process will be out of control for the $k +$ 1st item. Unfortunately, no easy method for computing the critical value $\zeta$ was given. Shiryaev (1963) studied a similar model but in continuous time and with a slightly different criterion. The continuous time process was a diffusion process and the operating characteristics of the control procedure could be found by solving a second order linear differential equation, thus enabling the appropriate critical value $\zeta$ to be found. Taylor (1967) applied the continuous time computations of Shiryaev to the discrete time model of Girshick and Rubin and plotted the optimal choice for $\zeta$ as a function of the costs and other system parameters. No proof of the validity of the continuous time approximations to the discrete time process was given, however. This paper fills this gap by exhibiting a sequence of Girshick and Rubin processes that converges to the continuous process of Shiryaev, and by showing that the corresponding cost functionals also converge. This procedure of obtaining a limiting continuous process from a sequence of discrete processes is very naturally carried out as an application of the theory of weak convergence of probability measures; cf. Billingsley (1968). More precisely, let $(X_i: i = 1,2,\cdots)$ be a sequence of independent Gaussian random variables with zero mean and unit variance. Let $S_0 = 0, S_k = X_1 + \cdots + X_k$ for $k = 1,2,\cdots$, and define for $n = 1,2,\cdots$ and $k = 0, 1,\cdots, n, X_k(k) = S_k/n^{\frac{1}{2}}$. Let $T_n$ be a geometric random variable with $P\lbrack T_n = j\rbrack = (\pi/n)(1 - \pi/n)^{j-1}$ for $j = 1,2,\cdots$, and $n = \lbrack\pi\rbrack + 1, \lbrack\pi\rbrack + 2,\cdots$. Define $\theta_n(k) = 0\quad\text{for} 0 \leqq k < T_n = n^{-1} (k - T_n + 1) \text{if} T_n \leqq k,$ where $k = 0, 1,\cdots, n$. Finally let $Y_n(k) = \theta_n(k) + X_n(k)$. One interprets $Y_n(k)$ as the cumulative sum up to item $k$ of observations in a truncated (at $n$) Girshick and Rubin process where $\mu_0 = 0, \mu_1 = 1/n, \sigma^2 = 1/n$ and the probability of leaving control between any two items is $\pi_n = \pi/n$. Let $U_n(k)$ be the posterior probability that the production process will be out-of-control for the production of the $(k + 1)$st item, assuming no repair is made. It is convenient to take as our control variable, $Z_n(k)$, the monotone function of $U_n(k)$ given by $Z_n(k) = U_n(k)/(1 - U_n(k))$. Then using the recursion for $U_n(k)$ (cf. Taylor (1967)) it is easy to show by induction that $Z_n(0) = 0$ and for $k = 1,2,\cdots, n$ $Z_n(k) = \pi_n(1 - \pi_n)^{-1} \sum^{k-1}_{j=0} (1 - \pi_n)^{-j} \exp \{Y_n(k) - Y_n(k - j) - j/2n\}.$ Since our analysis will treat probability measures on $C\lbrack 0, 1\rbrack$, the space of continuous, real-valued functions on [0,1] with the metric $(\rho)$ of uniform convergence, it is convenient to introduce continuous versions of the processes $\{X_n(k)\}, \{\theta_n(k)\}, \{Y_n(k)\}$, and $\{Z_n(k)\}$. We define $x_n(t) = X_n(k) + (nt - k)(X_n(k + 1) - X_n(k))$ where $kn^{-1} \leqq t \leqq (k + 1)n^{-1}, k = 0,1,\cdots, n - 1$, and $0 \leqq t \leqq 1$. Observe that $x_n(t) = X_n(k)$ for $t = k/n$ and is defined by linear interpolation for other values of $t$. In a similar manner define $\theta_n(t), y_n(t)$, and $z_n(t)$. Clearly, $y_n(t) = x_n(t) + \theta_n(t)$. Let $\zeta$ be a fixed positive number and $z \varepsilon C\rbrack 0, 1\rbrack$, then define $\tau(z) = \inf \{ t: 0 \leqq t \leqq 1, z(t) = \zeta\}$ if the set $\{t: 0 \leqq t \leqq 1, z(t) = \zeta\}$ is not empty and 1 otherwise. A cost functional whose expected value may be used to measure long run time average cost of operation plus repair (Taylor, (1967)) is given by $C_\zeta^{(n)}(z_n) = \sum^{\rbrack n\tau(z_n)\rbrack}_{k=0} \{c(n^{-1})z_n(k/n) (1 + z_n(k/n))^{-1} - \gamma/n\}$ where $\gamma$ is a constant related to $K$, the repair cost. For the continuous version of Shiryaev (1963) let $\{x(t): 0 \leqq t \leqq 1\}$ be Brownian motion and $T$, an exponential random variable, independent of $\{x(t): 0 \leqq t \leqq 1\}$, with density $\pi e^{-\pi t}, t > 0$. Define the continuous time process $(\theta(t); 0 \leqq t \leqq 1)$ by \begin{align*}\theta(t) = 0\quad\text{if} t \leqq T \\ = t - T \text{if} t > T.\end{align*} Let $y(t) = \theta(t) + x(t)$. The control variable here is $z(t) = \pi \int^t_0 \exp \{\pi s + y(t) - y(t - s) - s/2\} ds$ (Shiryaev (1963)), and the cost functional is $C_\zeta(z) = \int^{\tau(z)}_0 \lbrack cz(s)(1 + z(s))^{-1} - \gamma\rbrack ds$. This is the process whose operating characteristics concerning $C_\zeta(z)$ are computed in Taylor (1967) and assumed to hold approximately for the discrete time process. Our principal objective in this paper is to establish in a rigorous fashion the manner in which the discrete processes converge to the continuous process. We show that the measures induced on $\mathscr{C}$, the Borel sets of $C\lbrack0, 1\rbrack$, by the processes $\{y_n(t): 0 \leqq t \leqq 1\}$ and $\{z_n(t): 0 \leqq t \leqq 1\}$ converge weakly as $n \rightarrow \infty$ to the measures induced by $\{y(t): 0 \leqq t \leqq 1\}$ and $\{z(t): 0 \leqq t \leqq 1\}$ respectively. As a consequence, we also show that the distributions of $\tau(z_n)$ and $C_\zeta^{(n)}(z_n)$ converge weakly to the distributions of $\tau(z)$ and $C_\zeta(z)$, respectively.