In the last decade there have been a number of papers dealing with the admissibility of translation invariant estimators of a location parameter. Blyth [2] treated sequential procedures in the case of normally or rectangularly distributed random variables. If $d$ is estimated and $\theta$ is the actual parameter value, for Blyth, op. cit., loss was measured by $W(|d - \theta|)$ where $W(\cdot)$ was a nondecreasing function on $\lbrack 0, \infty)$. In the same year Blackwell [1] treated the fixed sample size problem in the case of discrete random variables taking only a finite number of values. For Blackwell, op. cit., loss was measured by $W(d - \theta)$ where $W(\cdot)$ was assumed continuous and bounded from below but otherwise arbitrary. Blackwell showed that if the discrete random variables (which could be vector valued) took values only on the integer lattice points and if there was a unique minimax translation invariant estimator then it was admissible. Later papers by Karlin [7] and Stein [11] discuss the admissibility of Pitman's estimator for square error. In reviewing these results we discovered that if the loss satisfied \begin{equation*}\tag{0.1}0 \leqq x < y \text{implies} W(x) < W(y),\quad y < x \leqq 0 \text{implies} W(x) < W(y),\end{equation*} and if there were several minimax translation invariant estimators then no translation invariant estimator could be admissible. This and a related result constitutes Section 4. It was logical to ask the converse question, does uniqueness imply admissibility? In the case of square error Pitman's estimator (except for changes on sets of measure zero) is necessarily unique since the loss function is strictly convex. In the case of normally distributed random variables and symmetrical loss functions as considered by Blyth, op. cit., the sample mean is the uniquely determined minimax translation invariant estimator. In this paper we have restricted the discussion to random variables which have density functions relative to Lebesgue measure. Since it proved possible to deal with the question of admissibility for a larger class of estimators than the translation invariant estimators, we define generalized Bayes estimators as the solution of a minimization problem. Section 2 deals with the question of the existence of measurable solutions to the minimization problem. Sections 5 and 6 deal with the question, does uniqueness imply admissibility? Section 3 deals with the question of whether generalized Bayes estimators are strongly consistent and shows that under mild restrictions this is the case. Section 8 is a generalization of the results of Katz [8] for minimax estimators of $\theta \varepsilon \lbrack 0, \infty)$. We show how to construct such estimators whenever loss is measured by $W(d - \theta), W(\cdot)$ strictly convex, non-negative, $W(0) = 0$. Blyth, op. cit., proved admissibility only within the class of continuous risk functions. In Section 9 we remove this restriction. We then show that if the loss function $W(\cdot)$ is strictly convex and symmetrical then the sample mean based on $n$ observations is admissible, in the nonparametric context of estimating the mean of an unknown density function, within the class of all sequential procedures having expected sample size $\leqq n$.

Let $B$ be the $\sigma$-algebra of all Borel subsets of the real line $\mathscr{X}$ and $\mathscr{A}$ be a $\sigma$-algebra of subsets of a set $\mathscr{Y}$. Let $\nu$ be a probability measure on $\mathscr{A}$. Let $P$ be a $B \times \mathscr{A}$ measurable function on $\mathscr{X} \times \mathscr{Y}$ such that $P(\cdot, y)$ is a distribution function for each $y \varepsilon \mathscr{Y}$. As usual, $B \times \mathscr{A}$ is the $\sigma$-algebra generated by sets $C \times A$ for $C \varepsilon B$ and $A \varepsilon \mathscr{A}$. We observe $(X, Y)$ where $\nu$ is the marginal distribution of $Y$ and, for some unknown $\theta$, the conditional distribution function of $X - \theta$ given $Y$ is $P(\cdot, Y)$. It is desired to estimate $\theta$ with loss function \begin{equation*}\tag{1.1}L(\theta, d)\begin{align*} = a(\theta - d) \quad\text{if} d \leqq \theta\\ = b(d - \theta)\quad\text{if} d \geqq \theta.\end{align*}\end{equation*} For any statistical problem, let $\rho(\theta, \delta)$ be the risk when $\delta$ is the decision procedure used and $\theta$ is the value of the parameter. In all that follows, $\mu$ will be Lebesgue measure on the real line. In a problem involving a real parameter, we say that $\delta$ is almost admissible provided that, given any other procedure $\delta'$, if $\rho(\theta, \delta') \leqq \rho(\theta, \delta)$ for all $\theta$, then $\rho(\theta, \delta') = \rho(\theta, \delta)$ a.e. ($\mu$). The purpose of the present paper is to prove the THEOREM. Suppose that, in addition to the above assumptions, (i) for each $y \varepsilon \mathscr{Y}$, the unique $(1 - \alpha)$th quantile of $P(\cdot, y)$ is 0 where $\alpha = a/(a + b)$ and \begin{equation*}\tag{(ii)}\int dv(y) \int x^2 d_xP(x, y) < \infty.\end{equation*} Then, under the loss function given in (1.1), $X$ is an almost admissible estimate of $\theta\cdot$ The proof of the theorem will be given in Section 2. Section 3 contains the proof of the COROLLARY. If, in addition to the assumptions of the theorem, $P(\cdot, y)$ is either absolutely continuous for all $y \varepsilon \mathscr{Y}$ or has its points of increase in a fixed lattice for all $y \varepsilon \mathscr{Y}$, then $X$ is an admissible estimate of $\theta$. If $\theta$ has "uniform" a priori distribution on $\mathscr{X}$, then $X$ is the $\alpha$th quantile of the a posteriori distribution of $\theta$ given $X$. Hence, $X$ is the best invariant estimate of $\theta$. Farrell [3] has shown that when the unicity assumption in the theorem is violated $X$ cannot be almost admissible. The Theorem and Corollary are analogous to Theorems 1 and 2 of Stein [5] for the case of loss function $(\theta - d)^2$. In Stein's theorems (i) is replaced by $E(X \mid Y) = 0$ and (ii) is replaced by $\int d\nu(y)\lbrack \int x^2 d_xP(x, y)\rbrack^{\frac{1}{2}} < \infty$. Thus, Stein's result, as well as ours, requires one moment more than is intuitively necessary. In Stein's case, contrary to ours, the extra moment is not required if $\mathscr{Y}$ contains only one element. Similar results for the case of hypothesis testing were obtained by Lehmann and Stein [4] under the assumption of a first moment, again one more moment than is intuitively needed. The proofs in Sections 2 and 3 are similar to those used by Stein. The method is originally due to Blyth [2]. Since the passage from our corollary to the case of a sample of size $n$ is similar to Stein's, it will be omitted here. Section 4 contains examples which show that the conclusion of the theorem cannot be strict admissibility. These examples are analogous to Blackwell's [1] example for the case of the loss function $(\theta - d)^2$.

In this paper the problem of minimax point estimation of a function $g(\theta)$ of a parameter $\theta$ is considered, when the loss function is of the form $W(u(x), g(\theta)) = |u(x) - g(\theta)|^p, (p > 1)$ and $u(x)$ is an estimate with bounded risk. When Conditions A and B stated later hold, it is shown that a unique minimax estimate $u_0(x)$ exists and if $\{u_n(x)\}$ is any uniformly bounded minimax sequence then the risk functions of $u_n(x)$ converges uniformly to the risk function of $u_0(x)$, so that no almost subminimax estimate can exist which, though not a minimax estimate, has for a wide range of values of the parameter $\theta$, a lower value of the risk than that of the minimax estimate. Under some additional conditions, it is shown that an approximation to the minimax estimate $u_0(x)$ in the space $\mathscr{F}^{(p)}_\infty$ of functions with bounded risk, may be obtained by the minimax estimate $\bar{u}_N(x)$ in the finite dimensional linear space spanned by $N$ basis vectors $v_1, \cdots v_N$ of $\mathscr{F}^{(p)}_\infty$, so that the maximum risk of $\bar{u}_N(x)$ converges to that of $u_0(x)$. This may help in finding an approximation to a minimax estimate in non-standard problems, where it is difficult to guess a minimax estimate from invariance or other considerations and specially when the problem is a perturbation of a standard problem.

In Chapter 10 of his classical work on sequential analysis [8], Wald started a program for dealing with multiple decision problems based on sequential estimation, and this work was to some extent further developed by Stein [7]. However, as far as the present writer is aware, this program was never completed. In a recent paper [6] a sequential procedure for estimating the mean of a normal distribution was given and the results applied to the problem of deciding which of $k$ non-overlapping intervals contains the mean. In the present paper, the main results of [6] are first derived in a different manner, using a slight modification of a procedure given by Wald (see Chapter 10 of [8]). This new derivation lends itself to dealing with other situations, and sequential confidence limits are worked out explicitly for two other cases, namely, for the variance and for the ratio of variances of normal distributions. These results are then applied to get closed decision procedures for a number of decision problems, including (1) testing a hypothesis regarding the mean of a normal distribution against either one-sided or two-sided alternatives; (2) comparing the means of $k$ experimental categories with a standard or control; (3) testing a hypothesis about the variance of a normal distribution; (4) deciding which of $k$ non-overlapping intervals contains the variance; (5) testing a hypothesis about the ratio of variances; (6) comparing the variances of $k$ experimental categories with a standard or control. In addition, a brief discussion of "mixed" problems, where we are concerned with finding a confidence interval for the parameter after a decision has been reached is given in Sections 3.1 and 3.3. In some of these problems open sequential solutions (in which there is no upper bound to the number of measurements required to reach a decision) are already known which have some optimum properties. However, for administrative and other reasons it is often desirable to restrict consideration to closed sequential procedures, so as to have an upper bound to the time or cost of an experiment. This has recently been emphasized by Armitage in connection with medical applications [2]. All the sequential procedures of the present paper are closed, but the point of closure in Section 3 depends on $s^2$ if $\sigma^2$ is unknown.

Invariant tests of the hypothesis that $\mathbf\Sigma_1 = \Sigma_2$ are based on the characteristic roots of $S_1S^{-1}_2$, say $c_1 \geqq c_2 \geqq \cdots \geqq c_p$, where $\Sigma_1$ and $\Sigma_2$ and $\mathbf{S}_1$ and $\mathbf{S}_2$ are the population and sample covariance matrices, respectively, of two multivariate normal populations; the power of such a test depends on the characteristic roots of $\Sigma_1\Sigma^{-1}_2$. It is shown that the power function is an increasing function of each ordered root of $\Sigma_1\Sigma^{-1}_2$ if the acceptance region of the test has the property that if $(c_1, \cdots, c_p)$ is in the region then any point with coordinates not greater than these, respectively, is also in the region. Examples of such acceptance regions are given. For testing the hypothesis that $\Sigma = I$, a similar sufficient condition is derived for a test depending on the roots of a sample covariance matrix $\mathbf{S}$, based on observations from a normal distribution with covariance matrix $\Sigma$, to have the power function monotonically increasing in each root of $\Sigma$.

Yates (1939, 1940) suggested use of information about treatment differences contained in differences of block totals. The procedure given by Yates for three dimensional lattice designs (1939) and for balanced incomplete block (BIB) designs was adopted by Nair (1944) for partially balanced incomplete block (PBIB) designs and was later generalized by Rao (1947) for use with any incomplete block design. The procedure is called recovery of inter-block information and consists of the following stages. The method of least-squares is applied to both intra- and inter-block contrasts, assuming that the value of $\rho$, the ratio of the inter-block variance to the intra-block variance is known. This gives the so called "normal" equations for combined estimation. The equations involve $\rho$ which is estimated from the observations by equating the error sum of squares (intra-block) and the adjusted block sum of squares in the standard analysis of variance to their respective expected values. This estimate is substituted for $\rho$ in the normal equations and the combined estimates are obtained by solving these equations. A priori, the inter-block variance is expected to be larger than the intra-block variance and hence it is customary to use the above estimator of $\rho$, truncated at unity. The error sum of squares in the inter-block analysis has at times been used in place of the adjusted block sum of squares (Yates (1939) for three dimensional lattice designs, Graybill and Deal (1959) for BIB designs). If $\rho$ were known, the combined estimators would have all the good properties of least-squares estimates. Since only an estimate of $\rho$ is used, the properties of the combined estimators have to be critically examined. One would expect these to depend on the type of estimator of $\rho$ used. To use the combined estimator of a treatment contrast with confidence one would like to know if it is unbiased and if its variance is smaller than that of the corresponding intra-block estimator, uniformly in $\rho$. The question of unbiasedness has been examined by some authors. Graybill and Weeks (1959) showed that for a BIB design, the combined estimator of a treatment contrast based on the Yates' estimator of $\rho$ in its untruncated form is unbiased. Graybill and Seshadri (1960) proved the same with Yates' estimator of $\rho$ in its usual truncated form, again for BIB designs. Roy and Shah (1962) showed that for any incomplete block design, if the estimator of $\rho$ is the ratio of quadratic forms of a special type, the corresponding combined estimators of treatment contrasts are unbiased. The customary estimator of $\rho$ (as given by Yates (1939) and Rao (1947)) is of the above type and hence gives rise to unbiased combined estimators. The variance of the combined estimators has also been examined by some authors. Yates (1939) used the method of numerical integration to show that for a three dimensional lattice design with 27 treatments and with 6 replications or more, the combined estimator of a treatment contrast has variance smaller than that of the intra-block estimator, uniformly in $\rho$. For a BIB design for which the number of blocks exceeds the number of treatments by at least 10 (or by 9 if in addition, the number of degrees of freedom for intra-block error is not less than 18), Graybill and Deal (1959) used the exact expression for the variance to establish this property of the combined estimators. In both the cases, the estimator of $\rho$ is based on the inter-block error and thus differs from the usual one based on the adjusted block sum of squares. For BIB designs, Seshadri (1963) gave yet another estimator of $\rho$ which gives rise to more precise combined estimators provided that the number of treatments exceeds 8. Roy and Shah (1962) gave an expression for the variance of the combined estimator based on any estimator of $\rho$ belonging to the class described above. Shah (1964) used this expression to show that the combined estimator of any treatment contrast in any incomplete block design has variance smaller than that of the corresponding intra-block estimator if $\rho$ does not exceed 2. The question that now arises is whether a combined estimator for a treatment contrast can be constructed which is "uniformly better" than the intra-block estimator, in the sense of having a smaller variance for all values of $\rho$. It is shown in Section 4 that for a linked block (LB) design with 4 or 5 blocks, recovery of inter-block information by the Yates-Rao procedure may even result in loss of efficiency for large values of $\rho$. A method of constructing a certain estimator of $\rho$, applicable to any incomplete block design for which the association matrix has a nonzero latent root of multiplicity $p > 2$, is presented in Section 3. For any treatment contrast belonging to a sub-space associated with the multiple latent root, the combined estimator based on this estimator of $\rho$ is shown to be uniformly better than the intra-block estimator if and only if $(p - 4) \times (e_0 - 2) \geqq 8$, where $e_0$ is the number of degrees of freedom for error (inter-block). For almost all well-known designs, the association matrix has multiple latent roots and this method can therefore be applied to many of the standard designs, at least for some of the treatment contrasts. It may be noted that, in general, this estimator of $\rho$ is different from the customary one given by Yates (1939) and Rao (1947). For LB designs however, this estimator of $\rho$ coincides with the customary one. It is shown here that, for a LB design, the usual procedure of recovery of inter-block information gives uniformly better combined estimators for all treatment contrasts if the number of blocks exceeds 5. As was pointed out before, if the number of blocks is 4 or 5 and if $\rho$ is large, recovery of inter-block information by the usual procedure results in loss of efficiency. Using the above method, we obtain an estimator of $\rho$ which produces a combined estimator uniformly better than the intra-block estimator for any treatment contrast for the following designs: (i) a BIB design with more than five treatments (ii) a simple lattice design with sixteen treatments or more and (iii) a triple lattice design with nine treatments or more. Applications to some other two-associate partially balanced incomplete block designs and to inter- and intra-group balanced designs have also been worked out in Sections 4 and 5. A computational procedure for obtaining the estimate of $\rho$ has been given for each case.

The vector of medians $\mathbf{M}_n$ and the vector of medians of averages of pairs $\mathbf{W}_n$ are investigated as competitors of the vector mean $\dot{\mathbf{X}}_n$ in estimating the location parameters in the $p$-variate one-sample problem. These estimates are found to be asymptotically normal and unbiased. Necessary and sufficient conditions for the degeneracy of the asymptotic distribution of $\mathbf{M}_n$ and $\mathbf{W}_n$ are given. For $\mathbf{W}_n$, in the case $p = 2$, these reduce to the condition that one coordinate variable be a monotone function of the other. Sufficient symmetry conditions are given for the asymptotic independence of the coordinates of these estimates. $\mathbf{W}_n$ and $\mathbf{M}_n$ when compared to $\dot{\mathbf{X}}_n$ in terms of the Wilks generalized variance are robust in the case of asymptotically independent coordinates. But for $p \geqq 3$ they can have arbitrarily small efficiency even in the non-singular $p$-variate normal case, if the underlying distribution is permitted to approach a suitable degenerate distribution arbitrarily closely. For $p = 2$, in the normal case, $\mathbf{W}_n$ is highly efficient, although $\mathbf{M}_n$ can have arbitrarily small efficiency. However, $\mathbf{W}_n$ is also shown to have arbitrarily small efficiency for a suitable highly correlated family of distributions even in the case $p = 2$. On the other hand, $\mathbf{W}_n$ becomes infinitely more efficient than $\dot{\mathbf{X}}_n$ as a given fixed distribution is mixed with an increasingly heavy gross error distribution. The behavior of these estimates is also considered for other non-normal families.

The hypothesis that the two independent random samples $u_1, \cdots, u_m$ and $v_1, \cdots, v_n$ come from the same (unknown) continuous distribution may be tested by the two-sample Cramer-von Mises criterion $W^2 = mn(m + n)^{-2} \sum d^2_i,$ or by $U^2 = mn(m + n)^{-2} \sum (d_i - \bar{d})^2,$ as proposed by Watson [4]. Here $d_i$ is the difference between the sample distribution functions at the $i$th point in the pooled sample; more precisely, if there are $m_i$ members of the first sample and $n_i$ members of the second sample contained in the first $i$ members of the pooled set of $m + n$ members arranged in order of magnitude, then $d_i = n_i/n - m_i/m,\quad i = 1, 2, \cdots, m + n;$ and $\bar{d} = \sum d_i/(m + n)$. Watson's $U^2$ is particularly appropriate when the sample members lie on a circle with no preferred initial point. The limiting distributions of $W^2$ and $U^2$ for large $m, n$ are known [1], [4]. Comparisons between the upper tails of the limiting distributions and small-sample distributions have been made by Anderson [2] for $W^2$ with $m, n \leqq 7$ and $m = n = 8$; by the writer [3] for $W^2$ with $m = n \leqq 12$; and by Watson [5] for $U^2$ and $W^2$ with $m = n = 10$. In this paper similar comparisons are made, for both $U^2$ and $W^2$, for all sample pairs with $m, n \geqq 4$ and $m + n \leqq 17$. For each such $m, n$, the true probability $P$ of attaining or exceeding $U^2$ is tabulated for every attainable $U^2$ with $P \leqq 0.1$. The correction factor $R = P/P_\infty$ is also given, where $P_\infty$ is the approximation to $P$ obtained by using the limiting distribution. For $W^2$, the values of $P$ and $R$ are tabulated for the smallest $W^2$ significant at each of the levels $(10, 8, 6, 5, 4, 3, 2.5, 2, 1.5) \times 10^{-a},\quad a = 2, 3, 4,$ except for the largest and second largest values of $W^2$, for which special formulae are supplied. It is found that for all $U^2$ and $W^2$ other than the largest and second largest values of each, when $P$ lies between 0.1 and 0.01, $R$ lies between 0.65 and 1.35 (except when $m = n = 5$), and when $P$ lies between 0.01 and 0.001, $R$ lies between 0.47 and 1.02. Less precisely, we may sum up by saying that the limiting distribution yields upper-tail areas correct within $\pm 35$ per cent between the 10 per cent and 1 per cent levels, but it overestimates the true $P$ by a factor between 1 and 2 between the 1 per cent and 0.1 per cent levels.

In this paper, a two-sample, two-stage nonparametric estimation problem will be studied. The parameter $\theta = \theta(F, G)$ under consideration is estimable (i.e., there exists an unbiased estimator $\phi = \phi(X_1, \cdots, X_r; Y_1, \cdots, Y_s)$ of $\theta$). $\phi$ is a function of independent observations from two populations with cumulative distribution functions $F(X)$ and $G(Y)$. The functions $F(X)$ and $G(Y)$ belong to a specified class $D$, such that a $U$-statistic based on $\phi$ is the unique minimum variance unbiased estimator of $\theta$. The total number of observations on populations $X$ and $Y$ will be a fixed number $N$. The sampling procedure is carried out in two stages. First, take $M$ observations from each of the populations; then allocate the remaining $N - 2M$ observations between the populations. The method of allocation utilizes the information from the first stage observations. Two kinds of two-stage estimators, represented by $U'$, and $U'$ will be introduced in this paper. Both $U'$ and $U"$ are $U$-statistics with random sample sizes. $U'$ is based essentially on the second stage observations only. $U''$ is defined on all $N$ observations. Intuitively, the statistic $U''$ is more appealing. The first stage observations are used not only to determine the allocation of the second stage observations, but also to estimate the parameter $\theta$. (see Section 3) One of the main results (Section 4) is that $U'$ is unbiased and under certain conditions, the variance of $U'$ approaches asymptotically to a particular variance $V_0$. (Here we shall consider the cases that both the variances of $U'$ and $U''$, are finite.) $U''$ is in general biased. However, under the same conditions the value $E(U'' - \theta)^2$ approaches asymptotically to the same value $V_0$. This value $V_0$ is the smallest variance of any one-stage $U$-statistic estimator of $\theta$, subject to the restriction that the total number of observations on $X$ and on $Y$ is $N. V_0$ is computed (see Section 2) when the best one-stage allocation of $N$ observations to the two populations is made with the help of partial or even complete information about the distributions $F(X)$ and $G(Y)$. Such information about $F$ and $G$ is represented by the "nuisance parameters" $b_{10} = b_{10}(F, G), b_{01} = b_{01}(F, G)$, etc., defined in Section 2. No prior knowledge of $b_{10}$ and $b_{01}$ is required to compute $\operatorname{Var}(U')$, and $E(U'' - \theta)^2$. In Section 5, the "optimal" choice of the first stage sample size $M$ relative to the fixed total sample size $N$ is discussed. The term "optimal" is in the sense that the particular choices of $M$ in relative order of magnitude of $N$, such that as $N$ goes to infinity, the ratios $\operatorname{Var}(U')/V_0$ and $E(U'' - \theta)^2/V_0$ approach unity as fast as possible in order of magnitude of $N$. Three cases with different conditions on $\phi$ are considered. It is found that the "optimal" choices depend on the specific conditions. Section 6 contains some examples. To each $\theta(F, G)$, the corresponding estimators for $b_{10}$ and $b_{01}$ together with their behavior under different conditions on $F$ and $G$, will be given. The examples include cases where the proposed procedures can be applied as well as cases where it cannot be applied. Section 7 shows the asymptotic normality of $U'$ and $U''$. Section 8 indicates that the proposed procedures can be extended to $k$-sample case, for $k > 2$, with similar results. The technique of two-stage estimation has been used in several papers. Stein [12] used it to determine confidence interval of a pre-assigned length for the mean of a normal population with unknown variance. Putter [8] used it to estimate the mean of a stratified normal population, Robbins [10] discussed such a technic for the design of experiments. Later, Ghurye and Robbins [4] used it to estimate the difference between the means of two normal populations (or some other specified populations). Richter [9] discussed the estimation of the common mean of two normal populations. During the preparation of the present paper, Alam [1] discussed the estimation of the common mean of $k \geqq 2$ normal populations. This paper generalizes these two-stage procedures in two ways. First, the underlying cumulative distributions $F, G$ are members of a larger class of distributions. Secondly, the underlying parameters $\theta(F, G)$ are not restricted to population means or functions of means. Consequently, in such a general setup the question of "the best" estimator of any particular parameter $\theta(F, G)$ is not considered in this paper.

In this paper we shall consider mathematical models motivated by chemical process control problems in which we wish to control the process to hold some property of the output material as nearly constant as possible. For example, we might control the viscosity of the process output by varying the setting of a valve in the cooling water line to a heat exchanger. We shall suppose that the viscosity is observed at equally spaced times and that these observations form a sequence $Y_1, Y_2, Y_3, \cdots$. The control procedure specifies the valve setting $X_1$ to be used during the time interval $(0, 1\rbrack$. Then, immediately after observing $Y_n$, we reset the cooling water valve to the position $X_{n+1}$ in accordance with the control procedure. The value of $X_{n+1}$ may depend on the prior values $X_1, \cdots, X_n$ and $Y_1, \cdots, Y_n$. In choosing $X_{n+1}$ we hope to hold $Y_{n+1}$ as close to a fixed value $Y_0$ as possible. More specifically, our objective will be to keep values of the loss $E(Y_n - Y_0)^2$ small. In this paper it is assumed that time lags attributable to process dynamics are negligible when compared with the interval between observations. Parts of the theory are being generalized to allow for the presence of such time lags and no serious difficulties have as yet been encountered in this type of generalization. The basic process and the general control procedure are described in mathematical terms in Sections 2 and 3. The concept of a system consisting of the process and the control procedure is then introduced in Section 4. The specific type of control procedure we shall consider is a generalization of the simple proportional control procedure discussed by Box and Jenkins [1]. This procedure requires that we specify $X_1$ and then recursively specify $X_{n+1} = X_n - a_n(Y_n - Y_0)$. We may place bounds on $X_n$ as in (8) below. The sequence $\{a_n\}$ is a sequence of positive numbers which may be specified a priori as in Section 5 or sequentially as in Section 9. In Sections 6 and 7 we study the performance of certain processes when the sequence $\{a_n\}$ is specified a priori. Most important are cases where $a_n$ converges to a small value $a$ near zero. With the class of processes used in this paper, it turns out that this procedure for choosing $X_n$ is quite similar to the stochastic approximation procedure first presented by Robbins and Monro in 1951 [7] and later generalized by Dvoretzky [5] and other authors. A comprehensive review of work on stochastic approximation is contained in a recent article by Schmetterer [8]. In Section 8 we specialize to the study of certain stationary processes. Attention is focused on the asymptotic performance of the controlled process as a function of the limit $a$ of the sequence $\{a_n\}$ in our control procedure. Of particular interest is the asymptotic loss given by $\lim_{n\rightarrow\infty} E(Y_n - Y_0)^2$. A procedure for the sequential choice of the sequence $\{a_n\}$ is proposed in Section 9 and its theoretical properties are studied in Section 10. Conditions are given under which both $a_n$ and $E(Y_n - Y_0)^2$ converge. It turns out that this procedure for the sequential choice of $a_n$ is also a stochastic approximation procedure of the Robbins-Monro type. Numerical examples given in Section 11 indicate that the asymptotic loss obtained by using this procedure for sequentially choosing $a_n$ may be nearly as small as the asymptotic loss obtained when the sequence $\{a_n\}$ is chosen a priori to converge to the value of $a$ that minimizes the asymptotic loss. However, when $a_n$ is chosen sequentially, we need much less a priori knowledge of the process.

The problem of obtaining a satisfactory measure of association between two random variables is closely allied to that of obtaining a measure of the amount of information about one contained in the other. For the more closely associated are the random variables the more information about one ought to be given by an observation on the other and vice versa. It is not, therefore, surprising to find that there have been several suggestions for basing coefficients of association on the now celebrated measure of information introduced by Shannon [9] in the context of communication theory. (See Bell [1] for certain of these and for references to others). Now Shannon's measure of information was based on the notion of entropy which seems to be much more meaningful for finite probability spaces than it is for infinite spaces, and while Gel'fand and Yaglom [2] have suggested a generalisation of Shannon's measure for infinite spaces, there remain difficulties, as indicated by Bell [1], about deriving from it coefficients of association or dependence between random variables taking infinite sets of values. In the present paper, by adopting a slightly different attitude to information from that of communication theory, we shall obtain a general measure of information which yields a fairly natural coefficient of dependence between two continuous random variables or, more generally, between two non-atomic measures. The next section provides the motivation for the introduction of this measure of information and a general definition is given in Section 3. In Section 4 we discuss some of the properties of this measure regarded as a coefficient of association along the lines suggested by Renyi [8]. Finally, in Section 5, we indicate the relevance of this measure to estimation theory.

The purpose of this paper is to present formulae and to examine fundamental properties of polykays of deviates which are here called $d$-statistics. In particular, formulae for $d$-statistics in terms of polykays having no unit subscripts are presented, relations involving the $d$-statistics are derived, and application is made to finite moment formulae involving the sample mean.

Wishart has demonstrated and Kendall has justified the application of a combinatorial method to products of $k$-statistics. Wishart's combinatorial method is, essentially, that introduced by Fisher though modifications were necessary since Fisher's method is applied directly to the writing of cumulant results while Wishart's method applies to products of $k$-statistics. It is the purpose of this paper to show that the combinatorial method may be further modified and extended so as to produce products of polykays and to present general formulae resulting from the new combinatorial method.

Elementary symmetric functions (e.s.f.'s) of the roots of certain determinantal equations are often associated with test statistics in multivariate analysis. The $T^2$ of Hotelling (1951) is related to the first e.s.f. of the roots (this is also known as "sum of the roots" or "trace" of its associated matrix). The $\lambda$-criterion of Wilks (1932) appropriate to $k$ samples is related to the last e.s.f. of the roots (this is also known as "product of the roots"). Except for these two e.s.f.'s of the roots, very little is known in current literature of the other e.s.f.'s. Non-symmetric functions, like the largest or smallest root of Roy (1957), also arise in certain situations where their use are preferable to other known statistics. This paper is not concerned with symmetric or non-symmetric functions as test statistics. It is concerned with a unified treatment of the distribution problem of the e.s.f. of the roots. It begins with the derivation of the joint distribution of the e.s.f. of the roots of a multivariate matrix under null hypothesis (Section 2). The next three sections concern with the moment problem of the distribution. Section 3 cites previous material necessary for obtaining determinantal expressions of the moments. Section 4 deals with the proof by construction of obtaining determinantal expressions for moments and product-moments which is applied (Section 6) to the third moment of the second e.s.f. and the product-moments of the first, second and third e.s.f. for illustration purposes. Section 5 gives an evaluating formula for the determinantal expressions.

This paper continues a study, initiated in [1], of the stochastic independence of $V = X - Y$ and the mean of order $\gamma$, \begin{equation*}\tag{1}M_\gamma\begin{align*} = (1/\gamma) \log \lbrack(e^{\gamma X} + e^{\gamma Y})/2\rbrack\quad\text{for} \gamma \neq 0 \text{and} \gamma \neq \pm \infty\\ = (X + Y)/2\quad\text{for} \gamma = 0\\ = \max (X, Y)\quad\text{for} \gamma = + \infty\\ = \min (X, Y)\quad \text{for} \gamma = - \infty\end{align*}\end{equation*} for independent $X$ and $Y$. The case $\gamma = 0$ is solved by the well known result, if, for independent $X$ and $Y$, the variables $X + Y$ and $X - Y$ are independent, then both $X$ and $Y$ have normal distributions with a common variance. The first proof of this result may be found in Kac [6]. (Although the statement of this result there is much weaker, Kac's proof applies exactly to the statement above.) A corresponding result for the gamma distribution, proved in restricted form by Hogg [5], and proved without restrictions by Lukacs [7], is as follows: if $Z_1$ and $Z_2$ are independent positive random variables for which $Z_1 + Z_2$ and $Z_1/Z_2$ are independent, then both $Z_1$ and $Z_2$ have gamma distributions with a common scale parameter. This result may be used to complete the study of the independence of $M_\gamma$ and $V$ for independent $X$ and $Y$ when $\gamma$ is finite and non-zero, by letting $Z_1 = \exp (\gamma X)$ and $Z_2 = \exp (\gamma Y)$. It is clear, then, from the result of Hogg and Lukacs that both $\exp (\gamma Y)$ and $\exp (\gamma Y)$ must have gamma distributions with a common scale parameter. The resulting distributions of $X$ and $Y$ have been studied in [1]. Therefore, of this study, there remain yet to be completely solved only the cases $\gamma = +\infty$ and $\gamma = - \infty$. These two cases are essentially the same since if $M_{+\infty}$ and $V$ are independent, then so are $\min (-X, -Y)$ and $V$. Thus, the negatives of the distributions for which $M_{+\infty}$ and $V$ are independent will yield independent $M_{-\infty}$ and $V$. We consider only the case $\gamma = -\infty$. This problem is considered in a previous paper, [2], in which the distributions were restricted to be discrete. The main result of that paper is as follows: if $X$ and $Y$ are independent, non-degenerate, discrete random variables, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then $X$ and $Y$ both have geometric distributions with common location and scale parameters, but possibly different geometric parameters. If the probability mass function of the geometric distribution is written as \begin{equation*}\tag{2}p(x) = (1 - p)p^{(x-\theta)/c}\quad\text{for} x = \theta, \theta + c, \theta + 2c, \cdots\end{equation*} then, the parameter $\theta$ is a location parameter, $c$ is a scale parameter, and $p$ is called the geometric parameter, $c > 0, 0 < p < 1$. In this paper, the distributions of $X$ and $Y$ are restricted to be absolutely continuous. This will result in a characterization of the so-called exponential distribution whose density is \begin{equation*}\tag{3}f(x) = (1/\sigma) \exp (-(x - \theta)/\sigma)\quad\text{if} x > \theta\end{equation*} and zero otherwise. The parameter $\theta$ is a location parameter, and the parameter $\sigma > 0$ is a scale parameter, which is also the standard deviation of the distribution. In these terms the main result of this paper may be stated as follows: if $X$ and $Y$ are independent random variables with absolutely continuous distributions, and if $U = \min (X, Y)$ and $V = X - Y$ are independent, then both $X$ and $Y$ have exponential distributions with a common location parameter but with possibly different scale parameters. This result then gives a characterization of the exponential distribution, since it is easy to show the converse, that if $X$ and $Y$ are independent and if each has an exponential distribution with a common location parameter but with possibly different scale parameters, then $U$ and $V$ are independent. For related results, the reader is referred to papers by Fisz [3] and Rogers [8].

In [1], [2] we construct for each integer $n \geqq 2$, a real-valued, bounded, uniformly continuous statistic defined on $R^n$, nondecreasing in each real argument, which is a minimal sufficient statistic for the family of all probability distributions defined on the Borel field $\beta^n$ in $R^n$ and dominated by Lebesgue measure $\lambda_n$. In this paper let $\{P_\theta\}$ be a family of probability distributions dominated by Lebesgue measure and defined on the restriction of $\beta^n$ to a Borel set $A \subset R^n$. Let $f = (f_1, \cdots, f_k)$ and $g = (g_{11}, \cdots, g_{1n_1}, \cdots, g_{k1}, \cdots g_{kn_k})$ be continuous sufficient statistics for $\{P_\theta\}$ defined on $A$, with $f_i$ and $g_{ij}$ real-valued. If there are $k$ functions $h_i:R^1 \rightarrow R^{n_i}, i = 1, \cdots, k$ so that $(g_{i1}, \cdots, g_{in_i}) = h_i \circ f_i$ a.e. $(\lambda_n)$, then is $g$ everywhere a continuous function of $f$, i.e., $g = h \circ f$ for continuous $h:f\lbrack A \rbrack \rightarrow g\lbrack A \rbrack$? If in addition $n_i = 1, i = 1, \cdots, k$ and each $h_i$ is a 1-1 function, are $f$ and $g$ identical, i.e., $g = h \circ f$ for bicontinuous $h:f\lbrack A \rbrack \rightarrow g\lbrack A \rbrack$? Now if (1) $A$ is connected, (2) $A$ has a dense interior, and (3) almost every linear section of each $f_i$ (and $g_i$ in the second case) satisfy Lusin's condition $(N)$, the answer to the above questions is affirmative (see Section 2 for definitions). But if at least one of (1), (2), or (3) is not satisfied, an affirmative answer is not in general possible (see Examples, Section 3). In Section 5 we show that this implies it is not possible to find a real-valued continuous minimal sufficient statistic $f$ defined on $R^n$ such that almost every linear section of $f$ satisfies Lusin's condition $(N)$, for some familiar probability distributions.

Results concerning the exponential convergence to zero of the probability of large deviations of the sample mean have already been obtained in the literature for instance, in Cramer [5], Chernoff [4], Bahadur and Rao [1], etc. We establish similar results about the exponential convergence to zero of the probability of large deviations of the sample distribution function, and more generally, of families of sample means.

The distribution of the number of successes in $n$ independent trials is "bell-shaped". The expected number of successes, $\mu$ say, either determines the most probable number of successes uniquely or restricts it to the pair of integers nearest to $\mu$.

This paper studies the present cost $C(\rho)$ of a renewal process, defined as the sum of the values of the costs of the replacements, considered at the starting time of the renewal process, with a compound interest. The characteristic function of $C(\rho)$ is found when the inter-arrival times $X_j$ are negatively exponentially distributed. The asymptotic properties of $C(\rho)$ as the force of interest $\rho$ tends to zero are studied in the general case, obtaining that if $X_1$ has finite moments of all orders, $C(\rho)$ is asymptotically normal; moreover, if we assume $EX^2_1 < \infty$, then the existence of all the moments of $X_1$ is necessary in order that the moments of $C(\rho)$ converge to the moments of the normal distribution.

In this paper we prove several asymptotic results about the expected length of time until first passage of a vector valued random walk. We suppose $X_i = (X^1_i, \cdots, X^k_i), i \geqq 1$, is a sequence of independently and identically distributed random vectors. We assume throughout that \begin{equation*}\tag{1}\max_{1 \leqq j \leqq k}E|X^j_1| < \infty; \mu_j = EX^j_1, 1 \leqq j \leqq k; \end{equation*} \\ \begin{equation*} \text{and that} 0 < \min_{1 \leqq j \leqq k} \mu_j.\end{equation*} We consider the following kind of first passage problem. Let $p(\cdot, \cdots, \cdot)$ be a real valued function of $k$ variables. Given $c > 0$ define a stopping variable $N(c) \geqq 1$ to be the least integer $n$ such that \begin{equation*}\tag{(2)}p \big (\sum^n_{i=1} X^1_i, \cdots, \sum^n_{i=1} X^k_i\big) > c,\end{equation*} with $N(c) = \infty$ if the inequality (2) fails for all $n \geqq 1$. The main result of this paper is: THEOREM 1. Suppose the function $p(\cdot, \cdots, \cdot)$ is a homogeneous function of degree one. Given the above assumptions, with probability one $\lim_{c\rightarrow\infty} c^{-1}N(c) = 1/p(\mu_1, \cdots, \mu_k)$. Suppose there is a real number $\alpha > 0$ such that \begin{equation*}\tag{3}\text{if} \min_{1 \leqq j \leqq k} a_j \geqq 0 \text{then} \alpha p(a_1, \cdots, a_k) \geqq \min_{1\leqq j \leqq k} a_j.\end{equation*} Then \begin{equation*}\tag{4} \lim_{c\rightarrow\infty}E|c^{-1} N(c) - 1/p(\mu_1, \cdots, \mu_k)| = 0.\end{equation*} If one views Theorem 1 in the light of Doob [2] Theorem 1 becomes very plausible. It is natural to define a continuous parameter process \begin{equation*}\tag{5}X(c) = c - p\big(\sum^{N(c) -1}_{i=1} X^1_i, \cdots, \sum^{N(c)-1}_{i=1} X^k_i\big).\end{equation*} In case the random variables $X^1_1, \cdots, X^k_1$ are nonnegative, using additional assumptions about $p(\cdot, \cdots, \cdot)$ it can be shown that asymptotically as $c \rightarrow \infty$ the process $X(\cdot)$ becomes a stationary Markov process. We plan to examine this question further in another place. The language of this paper is that of random variables and probability. We use freely the idea of stopped random walks. This makes it necessary to define, in the sequel, many different stopping variables. To simplify notation we use the same notation for different stopping variables and leave it to the context to distinguish the various random variables. Theorem 1 is a corollary of Theorem 3, stated below. The sequence of the argument is to prove a one-dimensional result, Theorem 2, then to prove by mathematical induction a $k$-dimensional result, Theorem 3, for the choice $p(x_1, \cdots, x_k) = \min_{1 \leqq j \leqq k} x_j$. The statements of Theorem 2 and Theorem 3 follow. THEOREM 2. Suppose $\{X_i, i \geqq 1\}$ is a sequence of independently and identically distributed random variables such that $E|X_1| < \infty \text{and} EX_1 = \mu > 0.$ If $c > 0$ is a real number define an integer valued random variable $N(c)$ to be the least integer $n \geqq 1$ such that $X_1 + \cdot + X_n > c$, with $N(c) = \infty$ if for all $n \geqq 1, X_1 + \cdot + X_n \leqq c$. Let $h(\cdot)$ be a real valued continuous strictly increasing convex function defined on $\lbrack 0, \infty)$ such that $h(0) = 0$ and $\lim_{x\rightarrow\infty} h(x)/x = \infty$. Assume that $Eh(|X_1|) \leqq \Delta < \infty$. Then $\mathrm{(6)}P(N(c) < \infty) = 1; EN(c) < \infty;$ $c \leqq \mu EN(c) \leqq c + h^{-1}(\Delta EN(c)).$ Convex functions of the type mentioned in the statement of Theorem 2 always exist. We discuss this question briefly at the beginning of Section 2. THEOREM 3. Suppose $\{X_i, i \geqq 1\}$ is a sequence of independently and identically distributed $k$-dimensional random vectors satisfying (1). Let $p(x_1, \cdots, x_k) = \min_{1 \leqq j \leqq k} x_j$ and let $N(c)$ be defined as in (2). Then $\mathrm{(7)}P(N(c) < \infty) = 1; EN(c) < \infty;$ $\lim_{c\rightarrow\infty} c^{-1}EN(c) = 1/(\min_{1 \leqq j \leqq k}\mu_j).$` We prove Theorem 2 in Section 2 for the sake of completeness and in order to have a result including a uniformity condition. Theorem 3 does not have any uniformity condition. Theorem 4 is a partial generalization of Theorem 2 in regard to a uniformity condition. THEOREM 4. Assume the hypotheses and definitions of Theorm 3. In addition assume that if $2 \leqq j \leqq k, i \geqq 1$, then $X^j_i \geqq 0$, and that $0 < \mu_1 \leqq \mu_2 \leqq \cdot \leqq \mu_k$. Let $h(\cdot)$ be a real valued nonnegative continuous function defined on $\lbrack 0, \infty)$ which satisfies \begin{equation*}\tag{(8)}h(0) = 0; \lim_{x\rightarrow\infty}h(x) /x = \infty.\end{equation*} There exists a function $a_k(\cdot, \cdot, \cdot)$ of three variables (and also depending on $h(\cdot)$ such that if $\mu > 0, \Delta > 0$ then $\lim_{c\rightarrow\infty} a_k(c, \mu, \Delta) = 0$ and if \begin{equation*}\tag{(9)}\max_{1 \leqq j \leqq k} Eh(|x^j_1|) \leqq \Delta, \min_{1 \leqq j \leqq k}EX^j_1 \geqq \mu_1,\end{equation*} then \begin{equation*}\tag{(10)}1/\mu_1 \leqq c^{-1}EN(c) \leqq 1/\mu_1 + a_k(c, \mu_1, \Delta).\end{equation*} The statement of Theorem 4 is asymmetric in that we require $X^2_1, \cdots, X^k_1$ to be nonnegative random variables. We believe that this restriction can be removed. But we have not been able to do so using the methods of this paper. The possibility of proving Theorems 2 and 4 (thereby obtaining results with uniformity conditions) was suggested to the author by arguments in a paper by Kiefer and Sacks [4]. Both Theorems have application to the study of the asymptotic behavior of the expected sample size in sequential tests. In regard to Theorem 2, since $\lim_{x\rightarrow\infty} h(x)/x = \infty$, if $d > 0$ then $\lim_{x\rightarrow\infty} h^{-1} (dx)/x = d \lim_{y\rightarrow\infty} h^{-1}(y)/y = d \lim_{z\rightarrow\infty} z/h(z) = 0$. Therefore from (6) it follows that $\lim_{c\rightarrow\infty} c^{-1} EN(c) = \mu^{-1}$. This part of Theorem 2 is known. The results of this paper on the length of time to first passage are related to the problems of renewal theory. We refer the reader to Smith [6] for a summary of results. The only multidimensional result, along the lines of this paper, in print, seems to be that of Chung [1]. In he proofs we use freely the idea of a stopped random walk and in connection with this an identity due to Wald [7]. Because this identity is used repeatedly we state it here. Suppose $\{Z_n, n \geqq 1\}$ is a sequence of independently and identically distributed random variables and $E|Z_1| < \infty$. Suppose $N$ is a random variable taking only positive integer values and $EN < \infty$. Suppose for each integer $n \geqq 1$ the event $\{N \geqq n\}$ is independent of the random variables $\{Z_m, m \geqq n\}$. Then $E\|Z_1 + \cdots + Z_N\| < \infty \text{and} E(Z_1 + \cdots + Z_N) = (EN)(EZ_1)$. Throughout the remainder of this paper we refer to this as Wald's identity and omit further mention of the reference. In the proofs given below many of the arguments require the definition of new stopping variables. For the sake of brevity the definitions of these stopping variables omit the statement that the value is $\infty$ if the stated condition fails to hold for any positive integer $n$. In this context the meaning of statements like "$P(N < \infty) = 1$" is that the exceptional set in question has measure zero. The proof of Theorem 2 is given in Section 2. The proofs of Theorem 3 and Theorem 4 require several lemmas proven in Section 3. The proof of Theorem 3 is given in Section 4, of Theorem 4 in Section 5, and of Theorem 1 in Section 6. The author is indebted to W. L. Smith for the present form of the statement of Lemma 3 and also for a very much shorter proof.

Let $\{x_{i1}, \cdots, x_{in_i}\}, i = 1, 2, \cdots, k, k \geqq 2, n_i \geqq 2$, be random samples from stochastically independent Gaussian populations with unknown means $\mu_i$ and unknown variances $\sigma^2_i$. To test the hypothesis $H_0:\sigma^2_1 = \cdots = \sigma^2_k = \sigma^2$, unknown, one may use a likelihood ratio test wherein $H_0$ is rejected if $\lambda < \lambda_0$, $\lambda = \prod^k_{i=1} \big\lbrack N \sum^{n_i}_{j=1} (x_{ij} - \bar{x}_i)^2/n_i \sum^k_{i=1}\sum^{n_i}_{j=1} (x_{ij} - \bar{x}_i)^2 \big\rbrack^{\frac{1}{2}n_i},$ $N = \sum^k_{i=1} n_i, \bar{x}_i = \sum^{n_i}_{j=1} x_{ij}/n_i$ and $\lambda_0$ is determined by the significance level. Under $H_0, Y_i = \sum^{n_i}_{j=1} (X_{ij} - \bar{X}_i)^2/\sigma^2$ are stochastically independent chi-square variables with $n_i - 1$ degrees of freedom (throughout capital letters denote random variables and the corresponding lower case letters values in their ranges). A general discussion of such derivations can be found in [2], pp. 183-195. The distribution of $\Lambda$ under $H_0$ is then obtainable from the extended form of the corollary below with $A_i = N/n_i, \beta_i = n_i/2, d_i = (n_i - 1)/2$. The corollary is a particularization of results obtained from a generalized gamma distribution, introduced by Stacy [3], with density \begin{equation*}\tag{1}f(y_i; a_i, d_i, p_i) = p_iy^{d_i-1}_i \exp \lbrack - (y_i/a_i)^{pi}\rbrack/a^{d_i}_i\Gamma(d_i/p_i)\end{equation*} for $y_i \geqq 0, a_i, d_i, p_i$ all positive. (Throughout, only the nonzero portions of densities will be indicated.) Inconveniently, formulas (4) and (6) below must be evaluated by numerical methods. Since the case $k > 3$ follows by induction from that for $k = 3$, only the cases $k = 2, 3$ will be given here.