William Kruskal [1], Howard Raiffa [2], J. L. Hodges, Jr. and E. L. Lehmann [4], have shown that in certain Neyman-Pearson type problems of testing a simple hypothesis against a simple alternative, determining the sample size by means of a chance device yields improvements over fixed sample size procedures. The purpose of this paper is not only to investigate the general problem of randomizing over fixed sample size tests of a simple hypothesis against a simple alternative, but also randomizing over other fixed sample size procedures in topics such as confidence interval estimation, the $k$-decision problem, etc. In Section 2, a fixed sample size test of a simple hypothesis against a simple alternative is identified with an operating characteristic $(\alpha, \beta, n)$ where $\alpha$ denotes the probability of a type I error, $\beta$ denotes the probability of a type II error, and $n$ denotes the sample size. A mixed single sample test is defined as a sequence of quadruples. $(\gamma_i, \alpha_i, \beta_i, n_i)$, where $\gamma_i \geqq 0, \sum^\infty_{i = 1}\gamma_i = 1$, where $(\alpha_i, \beta_i, \eta_i)$ is a fixed sample size test and where $\gamma_i$ is interpreted as the probability of using the fixed sample size test $(\alpha_i, \beta_i, n_i)$ for $i = 1, 2, \cdots$. A mixed single sample test is identified with an operating characteristic $(\alpha, \beta, n) = \sum^\infty_{i = 1}\gamma_i(\alpha_i, \beta_i, n_i)$. For each nonnegative integer $n$, the class $A_n$ of admissible fixed sample size procedures of sample size $n$ is defined in an obvious way. We define $A = \bigcup^\infty_{i = 0}A_i$ and $A^{\ast}$ as the convex hull of $A$. It is not necessarily true that $A^{\ast}$ is closed. An example is given to show this. However, it is true that the lower boundary of $A^{\ast}$ is a subset of $A^{\ast}$ so that the lower boundary of $A^{\ast}$ determines a minimally complete class, $\mathcal{a}$, of mixed single sample tests. The tests in $\mathcal{a}$ are characterized from a Bayes point of view and a technique for constructing the tests in $\mathcal{a}$ is given. In Section 3, the technique is applied to tests on the mean of a normal distribution with known variance. It is shown that the tests in $\mathcal{a}$ are either (a) fixed sample size tests, or (b) mixtures of at most two fixed sample size tests. It is shown that there exists a minimal subset $\mathcal{a}_0$ of $A$ such that all improved randomized procedures are of the form $(\alpha, \beta, n) = \gamma(0, 1, 0) + (1 - \gamma)(\alpha_0, \beta_0, n_0)$ or $(\alpha, \beta, n) = \gamma(1, 0, 0) + (1 - \gamma)(\alpha_0, \beta_0, n_0)$, where $0 < \gamma < 1$ and where $(\alpha_0, \beta_0, n_0) \varepsilon \mathcal{a}_0$. It is then shown how to construct $\mathcal{a}_0$. The following problems (of the Neyman-Pearson type) are solved: (a) Given $\alpha$ and $\beta$, how can we find the test in $\mathcal{a}$ with the given $\alpha$ and $\beta$? (b) Given $\alpha$ and $n$, how can we find the test in $\mathcal{a}$ with the given $\alpha$ and $n$? Numerical examples are worked out. In Section 4, the technique is applied to tests on the mean of a binomial distribution. Although no general results were obtained, numerical examples of interest are given. In Section 5, the technique is applied to tests on the range of a rectangular distribution (when one end point is known). It is shown that if $\alpha > 0, n > 0$, and $(\alpha, \beta, n) \varepsilon A_n$, then $(\alpha, \beta, n) \not\in \mathcal{a}$. The tests in $\mathcal{a}$ are characterized by a simple equation which makes it easy to (a) determine whether a given point $(\alpha, \beta, n)$ belongs to $\mathcal{a}$, and (b) construct any test in $\mathcal{a}$, given two of the three coordinates. It is shown that if $(\alpha, \beta, n) \varepsilon A_n$, then there exists a test $(\alpha, \beta, n')$ in $\mathcal{a}$ such that $n' = (1 - \alpha)n$. Hence, the fractional saving in the expected sample size achieved by randomization is equal to $\alpha$. In Section 6, it is shown that in tests on the mean of a rectangular distribution (with known range), it never pays to randomize. In Section 7, confidence intervals are evaluated in terms of confidence coefficient $(\alpha)$, expected length $(L)$ and expected sample size $(n)$. For the problem of obtaining a confidence interval for the mean of a normal distribution with known variance, "improved" randomized procedures exist and are of the form $(\alpha, L, n) = \gamma(0, 0, 0) + (1 - \gamma)(\alpha', L', n')$ where $0 < \gamma < 1$ and where $(\alpha', L', n')$ is a fixed sample size confidence interval procedure. Clearly, the randomized procedures obtained are of such a nature that the question of confidence intervals evaluated in terms of expected length and/or expected sample size is thrown open to discussion. In Section 8, the $k$-decision problem is discussed. It is shown that improvements can be obtained by randomization. In Section 9, the problem of applying mixed single sample tests of a composite hypothesis against a composite alternative is discussed. In Section 10, mixed single sample procedures are compared to Wald's sequential probability ratio test in the problem of tests on the range of a rectangular distribution when one endpoint is known and are shown to be efficient in a certain sense. In Section 11, the estimation problem is mentioned. It is shown that in most practical problems, fixed sample size procedures are optimal. In Section 12, applications of mixed single sample tests are discussed.

Let $X_1, \cdots, X_m$ and $Y_1, \cdots, Y_n$ be ordered observations from the absolutely continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. If $z_{Ni} = 1$ when the $i$th smallest of $N = m + n$ observations is an $X$ and $z_{Ni} = 0$ otherwise, then many nonparametric test statistics are of the form $$mT_N = \sum^N_{i = 1} E_{Ni}z_{Ni}.$$ Theorems of Wald and Wolfowitz, Noether, Hoeffding, Lehmann, Madow, and Dwass have given sufficient conditions for the asymptotic normality of $T_N$. In this paper we extend some of these results to cover more situations with $F \neq G$. In particular it is shown for all alternative hypotheses that the Fisher-Yates-Terry-Hoeffding $c_1$-statistic is asymptotically normal and the test for translation based on it is at least as efficient as the $t$-test.

The classical multivariate 2 sample significance test based on Hotelling's $T^2$ is undefined when the number $k$ of variables exceeds the number of within sample degrees of freedom available for estimation of variances and covariances. Addition of an a priori Euclidean metric to the affine $k$-space assumed by the classical method leads to an alternative approach to the same problem. A test statistic $F$ which is the ratio of 2 mean square distances is proposed and 3 methods of attaching a significance level to $F$ are described. The third method is considered in detail and leads to a "non-exact" significance test where the null hypothesis distribution of $F$ depends, in approximation, on a single unknown parameter $r$ for which an estimate must be substituted. Approximate distribution theory leads to 2 independent estimates of $r$ based on nearly sufficient statistics and these may be combined to yield a single estimate. A test of $F$ nominally at the 5% level but based on an estimate of $r$ rather than $r$ itself has a true significance level which is a function of $r$. This function is investigated and shown to be quite near 5%. The sensitivity of the test to a parameter measuring statistical distance between population means is discussed and it is shown that arbitrarily small differences in each individual variable can result in a detectable overall difference provided the number of variables (or, more precisely, $r$) can be made sufficiently large. This sensitivity discussion has stated implications for the a priori choice of metric in $k$-space. Finally a geometrical description of the case of large $r$ is presented.

For a family of exponential densities a method is given, called "$D$ method," for constructing a class of similar tests in the case that the minimal sufficient statistic is boundedly incomplete. This method also provides a proof of a criterion for bounded incompleteness. Under certain conditions the criterion states that a sufficient statistic for a family of exponential densities is boundedly incomplete if the number of components of the statistic is larger than the number of parameters specifying the distribution. Applications are indicated in the Behrens-Fisher problem, and in the problem of testing the ratio of mean to standard deviation in a normal population. In the latter problem it is shown that the $D$ method generates the whole class of similar tests. Some unsolved problems concerning the existence of an optimal similar test are indicated.

Various minimum $\chi^2$ methods used for generating B.A.N. estimates are summarized, and a new method which generates B.A.N. estimates as roots of certain linear forms is introduced and investigated. As a particular application of the method, the estimation of the bacterial density in an experiment using dilution series is considered.

This paper is concerned with non-sequential multiple-decision procedures for which the sample size is a minimum subject to either (1) lower bounds on the probabilities of making correct decisions or (2) upper bounds on the probabilities of making incorrect decisions. Such decision procedures are obtained by constructing artificial decision problems for which the minimax strategies provide solutions to problems (1) and (2). These are shown to be "likelihood ratio" and "unlikelihood ratio" decision rules, respectively. Thus, although problems (1) and (2) are formulated in the spirit of the classical Neyman-Pearson approach to two-decision problems, minimax theory is used as a tool for their solution. Problems of both "simple" and "composite" discrimination are considered and some examples indicated. (Some multivariate examples are given in [4].) Various properties of the decision rules are derived, and relationships with works of Wald, Lindley, Rao and others are cited.

Methods are given for constructing sets of simultaneous confidence intervals for the means of variables which follow a multivariate normal distribution. In section (3), a set of confidence intervals is obtained for each of two special cases; first when the variances are assumed to be known, and second when the variances are assumed to be equal. These two sets have the property that the confidence is known exactly, rather than merely being bounded below. In the case of known variances, the intervals are of fixed lengths (i.e., the lengths are the same from sample to sample); when the variances are unknown, the intervals are of variable lengths. It may be surprising to note that nothing need be known about the covariances in order to obtain confidence intervals of fixed lengths whose confidence coefficient is exact. These intervals are long, and do not make use of all the information provided by the sample. Each of sections (4) to (7) considers a different method for obtaining confidence intervals of bounded confidence level. In each section a set of fixed lengths is obtained when the variances are assumed to be known, while a set of variable lengths is obtained when the variances are unknown but equal. In section (5) the set of variable lengths applies to the general multivariate normal distribution, all the other confidence intervals in this paper require some assumption concerning the variances. In section (8) the sets of intervals are compared on the basis of length. One of the bounded confidence level methods, which has been established only for two or three variables or for an arbitrary number of variables with a special type of correlation matrix, is shown to yield the best possible set. Another of the bounded confidence level methods, whose use is established in general, is shown to be almost as good as the best set for confidence coefficients of practical interest. It is interesting to notice that intervals with bounded confidence level, are found which are much shorter than the ones whose confidence level is exact. This need not surprise us, however. In the case of just one variable, we might easily find hat the 95% confidence intervals for the mean using the $t$-statistic were shorter on the average than 94% confidence intervals using order statistics. Moreover, since in admitting sets of confidence intervals with bounded confidence level we consider a much broader class of methods, we might almost expect that some of them would give better intervals.

Let $\{X_1, X_2, \cdots, X_N\}$ be an observed sequence from a stochastic process, where $X_i$ can take any one of $s$ values $1, 2, \cdots, s$. Let $f_\mathfrak{u}$ be the frequency of the $m$-tuple $\mathfrak{u} = (u_1, u_2, \cdots, u_m)$ in the sequence. Let $H'_n$ be the composite hypothesis that the process is a Markov chain of order $n$. Let $H_n$ be any simple hypothesis belonging to $H'_n$. Let $H^{\ast}_n$ be the maximum likelihood $H_n$. Let the expected value of $f_\mathfrak{u}$ in a new sequence of length $N$ given $H_n$ be $f_{\mathfrak{u},n}$, and given $H^{\ast}_n$ be $f^{\ast}_{\mathfrak{u},n}$. Let $$\Psi^2_{m,n} = \sum_\mathfrak{u} (f_\mathfrak{u} - f_{\mathfrak{u},n})^2/f_{\mathfrak{u},n},$$ $$\Psi^{\ast 2}_{m,n} = \sum_\mathfrak{u} (f_\mathfrak{u} - f^{\ast}_{\mathfrak{u},n})^2/f^{\ast}_{\mathfrak{u},n},$$ $$\Psi^{\ast 2}_{n + 1,n} = 0.$$ Good had proposed in [7] the following two conjectures: (a) that the asymptotic distribution $(N \rightarrow \infty)$ of $\Psi^{\ast 2}_{m,n}$, when $H'_n$ is true, is $$\ast^{m - n - 1}_{\lambda = 1} K_{g(\lambda)} (x/\lambda),$$ where $\ast$ denotes convolution, $g(\lambda) = (s - 1)^2s^{m - 1 - \lambda}$, and $K_i(x)$ is the $\chi^2$-distribution with $i$ degrees of freedom; (b) that the asymptotic distribution of $\Psi^2_{m,n}$, when $H_n$ is true, is $$\ast^{m - 1}_{\lambda = 1} K_{g(\lambda)}(x/\lambda)\ast K_{s - 1}(x/m),$$ mathematically independent of $n$. Conjectures (a) and (b) were proved by Billingsley [2] for the special case $n = 0$. For the special case $n = -1$ (by convention, $H'_{-1}$ is the hypothesis of equiprobable or perfect randomness (see [7])), Conjecture (b) was proved by Good [5] when $s$ is prime. In the present paper, Conjecture (a) will be proved for the general case $n \geqq -1$; conjecture (b) will be shown to be incorrect for $n > 0$, although a modified version of (b) will be proved for $n \geqq -1$. A third conjecture by Good [6] will also be proved here. It was assumed in these earlier papers, and it will be assumed here, that all transition probabilities in the Markov chain are positive; the results can be modified accordingly when some of these probabilities are zero (see [1] and [10]). Let $M_{m,n} = -2 \log\lambda_{n,m - 1}$, where $\lambda_{n,m - 1}$ is the ratio of the maximum likelihood given $H'_n$ to that given $H'_{m - 1}$ (see [6]). For $m = n + 2$, the statistics $\Psi^{\ast 2}_{m,n}$ is asymptotically equivalent, when $H'_n$ is true, to the likelihood ratio statistic $M_{m,n}$. For $m > n + 2, \Psi^{\ast 2}_{m,n}$ is asymptotically equivalent, when $H'_n$ is true, to $\sum^{m - n -1}_{\lambda = 1}\lambda M_{m + 1 - \lambda, m - 1 - \lambda}$, while $M_{m,n}$ is asymptotically equivalent to $$\sum^{m - n - 1}_{\lambda = 1} M_{m + 1 - \lambda, m - 1 - \lambda}$$ (see [6], [10]). Thus, $\Psi^{\ast 2}_{m,n}$ corresponds asymptotically to a weighted sum of the likelihood ratio statistics $M_{n + 2,n}, M_{n + 3, n + 1}, \cdots, M_{m,m - 2}$, with the weights $m - n - 1, m - n - 2, \cdots, 1$, respectively, while $M_{m, n}$ weights these statistics equally (see [13] and reference to [13] in Section 4 herein). Let $L_{m,n} = -2 \log \mu_{n,m - 1}$, where $\mu_{n,m - 1}$ is the ratio of the likelihood given $H_n$ to the maximum likelihood given $H'_{m - 1}$. For $m - 1 = n = 0$, the statistic $\Psi^2_{m,n}$ is asymptotically equivalent, when $H_n$ is true, to $L_{m,n}$. For $m - 1 > n = 0, \Psi^2_{m,n}$ is asymptotically equivalent, when $H_n$ is true, to $$\sum^{m - 1}_{\lambda = 1} \lambda M_{m + 1 - \lambda, m - 1 - \lambda} + mL_{n + 1,n},$$ while $L_{m,n}$ is asymptotically equivalent to $\sum^{m - 1}_{\lambda = 1} M_{m + 1 - \lambda, m - 1 - \lambda} + L_{n + 1, n}$. For $n > 0$, the relation between $\Psi^2_{m,n}$ and the likelihood ratio statistics $L_{m,n}$ and $M_{m,n}$ is not so straightforward. However, a modification $\Psi^{'2}_{m,n}$ of $\Psi^2_{m,n}$ (see Section 6 herein) is asymptotically equivalent, when $H_n$ is true, to $L_{m,n}$ for $m = n + 1$, and to $\sum^{m - n -1}_{\lambda = 1} \lambda M_{m + 1 - \lambda, m - 1 - \lambda} + (m - n)L_{n + 1, n}$ for $m > n + 1$; while the likelihood ratio statistic $L_{m,n}$ is asymptotically equivalent to $$\sum^{m - n - 1}_{\lambda = 1} M_{m + 1 - \lambda, m - 1- \lambda} + L_{n + 1,n}.$$ In [10], the $m$-tuple $\mathfrak{u}$ was "split" into an $(m - n - 1)$-tuple, an $n$-tuple, and a 1-tuple; thus obtaining $s^n$ "contingency tables" $(n \geqq 0)$ each $s^{m - n - 1} \times s$ (see [10]). The statistic $M_{m,n}$ can be seen to be asymptotically equivalent to the sum of the "likelihood ratio statistics" (for testing "independence" in each table) for the $s^n$ tables, and the asymptotic distribution, when $H'_n$ is true, of $M_{m,n}$ will be $\chi^2$ with $s^n(s^{m - n - 1} - 1)(s - 1) = s^m - s^{m - 1} - s^{n + 1} + s^n$ degrees of freedom. It is also possible to "split" the $m$-tuple $\mathfrak{u}$ into an $(m - n - 1 - r)$-tuple, and $\mathbf{n}$-tuple, and a $(1 + r)$-tuple $(0 \leqq r \leqq m - n - 2)$; thus obtaining $s^n$ "contingency tables," each $s^{m - n - 1 - r} \times s^{1 + r}$ (see [10]). The sum $_rM_{m,n}$ of the likelihood ratio (or any equivalent goodness of fit) statistics for the $s^n$ tables will have an asymptotic mean value, when $H'_n$ is true, of $$s^n(s^{m - n - 1 - r} - 1)(s^{1 + r} - 1) = s^m - s^{m - r - 1} - s^{n + 1 + r} + s^n.$$ but the asymptotic distribution will not be $\chi^2$ unless $r = 0$ or $m - n - 2$. It can be seen, using the methods developed in the present paper, that the statistic $_rM_{m,n}$ will be asymptotically equivalent, when $H'_n$ is true, to $$\sum^{m - n - 1}_{\lambda = 1} h(\lambda)M_{m + 1 - \lambda, m - 1 - \lambda},$$ where \begin{equation*}h(\lambda) = \begin{cases}\lambda \text{for} 0 < \lambda \leqq v\\v \text{for} v \leqq \lambda \leqq m - n - v\\(m - n - \lambda) \text{for} m - n - v \leqq \lambda \leqq m - n - 1,\end{cases}\end{equation*} and $v = \min \lbrack r + 1, m - n - r - 1\rbrack$. Thus, the asymptotic distribution $(N \rightarrow \infty)$, of $_rM_{m,n}$ (or the corresponding asymptotically equivalent goodness of fit statistics), when $H'_n$ is true, is $$\ast^{m - n - 1}_{\lambda = 1} K_{g(\lambda)}\lbrack x/(h(\lambda))\rbrack.$$ This result generalizes the earlier published results concerning the asymptotic distribution of the likelihood ratio statistic $M_{m,n}$ (or the corresponding asymptotically equivalent goodness of fit statistics) for testing the null hypothesis $H'_n$ within $H'_{m - 1}$, since $_rM_{m,n}$ for $r = 0$ or $m - n - 2$ is asymptotically equivalent to $M_{m,n}$ (see [6], [10]). A proof of this result will not be given since the method of proof is quite similar to that presented here for the asymptotic distribution of $\Psi^{\ast 2}_{m,n}$.

Using the generalized variance as a criterion for the efficiency of estimation, the best choice of fixed variable values within an interval for estimating the coefficients of a polynomial regression curve of given degree is determined for the classical regression model. Using this same criterion, some results are obtained on the increased efficiency arising from doubling the number of equally spaced observation points (i) when the total interval is fixed and (ii) when the total interval is doubled. Measures of the increased efficiency are found for the classical regression model and for models based on a particular stationary stochastic process and a pure birth stochastic process.

The paper is divided into two parts: A. An elementary derivation of Bartlett's results on the distribution of the canonical correlation coefficients using exterior differential forms. Briefly, our method consists of taking the original multivariate normal distribution, transforming to the canonical correlations and other variables, and then integrating out these extraneous variables. B. A new method of calculating the conditional moments which appear in Bartlett's expansion of this distribution, based on the process of averaging over the orthogonal group. This method allows the calculation of moments of any order.

Significance testing, as described in most textbooks, consists in fixing a standard significance level $\alpha$ such as .01 or .05 and rejecting the hypothesis $\theta = \theta_0$ if a suitable statistic $Y$ exceeds $C$ where $P_{\theta_0}\{Y > C\} = \alpha$. Such a procedure controls the probability of false rejection (error of the first kind) at the desired level $\alpha$ but leaves the power of the test and hence the probability of an error of the second kind to the mercy of the experiment. It seems more natural when deciding on a significance level (and this suggestion is certainly not new) to take into account also what power can be achieved with the given experiment. In Section 3 a specific suggestion will be made as to how to balance $\alpha$ against the power $\beta$ obtainable against the alternatives of interest. The adoption of this or some similar rule for choosing a significance level has important consequences for the theory of testing composite hypotheses, where nuisance parameters are present. Since the quantity $\alpha$ is then potentially a function of the nuisance parameter $\vartheta$, the classical rule of a fixed significance level leads to the condition that the tests be exact or similar, that is, that $\alpha(\vartheta)$ equal the preassigned value $\alpha$ for all $\vartheta$. On the other hand, the power $\beta$ that can be attained against any alternative $\theta = \theta_1$ frequently depends on $\vartheta$. The requirement that $\alpha(\vartheta)$ and $\beta(\vartheta)$ be in a certain balance thus leads to tests which are not similar and hence do not agree with the standard solutions. To obtain a suitable setting for this discussion, we consider first a minimal complete class of tests for testing the hypothesis $H:\theta \leqq \theta_0$ in a multiparameter exponential family (Section 2). The proposed $\alpha, \beta$-relation is discussed in Section 3, and in Section 4 is applied to the exponential family. Section 5 gives some illustrations of the theory.

Test criteria for (i) multivariate analysis of variance, (ii) comparison of variance-covariance matrices, and (iii) multiple independence of groups of variates when the parent population is multivariate normal are usually derived either from the likelihood-ratio principle [6] or from the "union-intersection" principle [2]. An alternative procedure, called the "step-down" procedure, has been recently used by Roy and Bargmann [5] in devising a test for problem (iii). In this paper the step-down procedure is applied to problems (i) and (ii) in deriving new tests of significance and simultaneous confidence-bounds on a number of "deviation-parameters." The essential point of the step-down procedure in multivariate analysis is that the variates are supposed to be arranged in descending order of importance. The hypothesis concerning the multivariate distribution is then decomposed into a number of hypotheses--the first hypothesis concerning the marginal univariate distribution of the first variate, the second hypothesis concerning the conditional univariate distribution of the second variate given the first variate, the third hypothesis concerning the conditional univariate distribution of the third variate given the first two variates, and so on. For each of these component hypotheses concerning univariate distributions, well known test procedures with good properties are usually available, and these are made use of in testing the compound hypothesis on the multivariate distribution. The compound hypothesis is accepted if and only if each of the univariate hypotheses are accepted. It so turns out that the component univariate tests are independent, if the compound hypothesis is true. It is therefore possible to determine the level of significance of the compound test in terms of the levels of significance of the component univariate tests and to derive simultaneous confidence-bounds on certain meaningful parametric functions on the lines of [3] and [4]. The step-down procedure obviously is not invariant under a permutation of the variates and should be used only when the variates can be arranged on a priori grounds. Some advantages of the step-down procedure are (i) the procedure uses widely known statistics like the variance-ratio, (ii) the test is carried out in successive stages and if significance is established at a certain stage, one can stop at that stage and no further computations are needed, and (iii) it leads to simultaneous confidence-bounds on certain meaningful parametric functions. 1.1 Notations. The operator $\varepsilon$ applied to a matrix of random variables is used to generate the matrix of expected values of the corresponding random variables. The form of a matrix is denoted by a subscript; thus $A_{n \times m}$ indicates that the matrix $A$ has $n$ rows and $m$ columns. The maximum latent root of a square matrix $B$ is denoted by $\lambda_{\max}(B)$. Given a vector $a = (a_1, a_2, \cdots, a_t)'$ and a subset $T$ of the natural numbers $1, 2, \cdots, t$, say $T = (j_1, j_2, \cdots, j_u)$ where $j_1 < j_2 < \cdots j_u$, the notation $T\lbrack a\rbrack$ will be used to denote the positive quantity: $$T\lbrack a\rbrack = + \{a^2_{j_1} + a^2_{j_2} + \cdots + a^2_{j_t}\}^{1/2}.$$ $T\lbrack a\rbrack$ will be called the $T$-norm of $a$. Similarly, given a matrix $B_{t \times t}$, we shall write $B_{(T)}$ for the $u \times u$ submatrix formed by taking the $j_1$th, $j_2$th, $\cdots, j_u$th rows and columns of $B$. We shall call $B_{(T)}$ the $T$-submatrix of $B$.

Several authors have studied the discrete stochastic process $(x_t)$ in which the $x$'s are related by the stochastic difference equation \begin{equation*}\tag{1.1}x_t = \alpha x_{t - 1} + u_t, \quad t = 1,2, \cdots, T,\end{equation*} where the $u$'s are unobservable disturbances, independent and identically distributed with mean zero and variance $\sigma^2$, and $\alpha$ is an unknown parameter. The statistical problem is to find some appropriate function of the $x$'s as an estimator for $\alpha$ and examine its properties. We may rewrite (1.1) as \begin{equation*}\tag{1.2}x_t = u_t + \alpha u_{t - 1} + \cdots + \alpha^{t - 1}u_1 + \alpha^tx_0.\end{equation*} From (1.2) we see that the distribution of the successive $x$'s is not uniquely determined by that of the $u$'s alone. The distribution of $x_0$ must also be specified. Three distributions which have been proposed for $x_0$ are the following: (A) $x_0$ = a constant (with probability one), (B) $x_0$ is normally distributed with mean zero and variance $\sigma^2/(1 - \alpha^2)$, (C) $x_0 = x_T$. Distribution (B) is perhaps the most appealing from a physical point of view, since if $x_0$ has this distribution and if the $u$'s are normally distributed, then the process is stationary (e.g., see Koopmans [4]). However, there are several analytic difficulties which arise in the statistical treatment of this process. Distribution (C), the so-called circular distribution, has been proposed as an approximation to (B) and is much easier to analyze (e.g., see Dixon [2]). Distribution (A) has been studied extensively by Mann and Wald [5]. An interesting feature of distribution (A) is that $\alpha$ may assume any finite value, while for distributions (B) and (C) $\alpha$ must be between $-1$ and 1. From (1.2) we see that a process satisfying (1.1) and (A) has \begin{equation*}\tag{1.3}\operatorname{var}(x_t) = \sigma^2(1 + \alpha^2 + \cdots + \alpha^{2(t - 1)})\end{equation*} If $|\alpha| \geqq 1, \lim_{t = \infty} \operatorname{var}(x_t) = \infty$ and the process is said to be "explosive." Mann and Wald [5] considered only the case $|\alpha| < 1$. They showed that the least squares estimator for $\alpha$ is the serial correlation coefficient \begin{equation*}\tag{1.4}\hat\alpha = \frac{\sum x_t x_{t - 1}}{\sum x^2_{t - 1}}\end{equation*} and that (for $|\alpha| < 1$) this estimator is asymptotically normally distributed with mean $\alpha$ and variance $(1 - \alpha^2)/T$. Rubin [6] showed that the estimator $\hat\alpha$ is consistent (i.e., $\operatorname{plim} \hat\alpha = \alpha$) for all $\alpha$. In this paper the asymptotic distribution of $\hat\alpha$ will be studied under the assumption that the $u$'s are normally distributed. For $|\alpha| > 1$, it is shown that the asymptotic distribution of $\alpha$ is the Cauchy distribution. For $|\alpha| = 1$, a moment generating function is found, the inversion of which will yield the asymptotic distribution.

For finite-state indecomposable channels, Shannon's basic theorem, that transmission is possible at any rate less than channel capacity but not at any greater rate, is proved. A necessary and sufficient condition for indecomposability, from which it follows that every channel with finite memory is indecomposable, is given. An important tool is a modification, for some processes which are not quite stationary, of theorems of McMillan and Breiman on probabilities of long sequences in ergodic processes.

In the discrete finite dam model due to Moran, the storage process $\{Z_t\}$ is known to be a Markov chain. Stationary distributions of $Z_t$ are obtained for the cases where the release is a unit amount of water per unit time, and the input is of (i) geometric, (ii) negative binomial and (iii) Poisson type. The paper concludes with a discussion of the problem of emptiness in the finite dam and considers the probability that, starting with an arbitrary storage, the dam becomes empty before it overflows.

When estimating the coefficients in a linear regression it is usually assumed that the covariances of the observations on the dependent variable are known up to multiplication by some common positive number, say $c$, which is unknown. If this number $c$ is known to be less than some number $k$, and if the set of possible distributions of the dependent variable includes "enough" normal distributions (in a sense to be specified later) then the minimum variance linear unbiased estimators of the regression coefficients (see [1]) are minimax among the set of all estimators; furthermore these minimax estimators are independent of the value of $k$. (The risk for any estimator is here taken to be the expected square of the error.) This fact is closely related to a theorem of Hodges and Lehmann ([3], Theorem 6.5), stating that if the observations on the dependent variable are assumed to be independent, with variances not greater than $k$, then the minimum variance linear estimators corresponding to the assumption of equal variances are minimax. For example, if a number of observations are assumed to be independent, with common (unknown) mean, and common (unknown) variance that is less than $k$; and if, for every possible value of the mean, the set of possible distributions of the observations includes the normal distribution with that mean and with variance equal to $k$; then the sample mean is the minimax estimator of the mean of the distribution. The assumption of independence with common unknown variance is, of course, essentially no less general than the assumption that the covariances are known up to multiplication by some common positive number, since the latter situation can be reduced to the former by a suitable rotation of the coordinate axes (provided that the original matrix of covariances is non-singular). This note considers the problem of minimax estimation, in the general "linear regression" framework, when less is known about the covariances of the observations on the "dependent variable" than in the traditional situation just described. For example, one might not be sure that these observations are independent, nor feel justified in assuming any other specific covariance structure. It is immediately clear that, from a minimax point of view, one cannot get along without any prior information at all about the covariances, for in that case the risk of every estimator is unbounded. In practice, however, one is typically willing to grant that the covariances are bounded somehow, but one may not have a very precise idea of the nature of the bound. One is therefore led to look for different ways of bounding the covariances, in the hope that the minimax estimators are not too sensitive to the bound. Unfortunately, in the directions explored here, the minimax estimator is sensitive to the "form" of the bound, although once the form has been chosen the minimax estimator does not depend on the "magnitude" of the bound. This result thus provides an instance in which the minimax principle is not too effective against the difficulties due to vagueness of the statistical assumptions of a problem, although this is a type of situation in which it has often been successful (see Savage in [4], pp. 168-9). In this note, two ways of bounding the covariances are considered. The first is equivalent to choosing a coordinate system for the "dependent variables," and placing a bound on the characteristic roots of the matrix of covariances of the coordinates, in terms of one of a certain class of metrics (e.g., placing a bound on the trace on the covariance matrix, or on its largest characteristic root). The second way consists of choosing a coordinate system, and then placing a bound on the variance of each coordinate. In the first situation, the minimum variance linear unbiased estimator corresponding to the case of uncorrelated coordinates, with equal variances, turns out to be minimax; this minimax estimator is, in general, different for different choices of coordinate system, but does not depend on the "magnitude" of the bound. Also, the minimax loss typically decreases at the rate of the reciprocal of the sample size. In the second situation, the minimax procedures derived here involve ignoring most of the observations, and applying a linear unbiased estimator to the rest. Again, the minimax procedure depends upon the choice of coordinate system; furthermore, in this case the minimax loss typically either does not approach zero with increasing sample size, or does so much more slowly than the reciprocal of the sample size. Thus the minimax estimator appears to be less unsatisfactory in the first situation than in the second, but in both cases it depends upon the choice of coordinate system, which is a disadvantage if there is no "natural" coordinate system intrinsic to the regression problem being considered. Section 2 below presents the formulation of the problem, and a basic lemma. Sections 3 and 4 explore the two ways of bounding the covariances just mentioned. Some examples are given in Section 5. I am indebted to R. R. Bahadur, L. J. Savage, and G. Debreu for their helpful comments.

In this paper we will study the effects on the covariance matrix of the least-squares estimates of regression coefficients and on the estimate of the residual variance when the usual condition of independence of residuals is violated. The cases of linear trend and of regression on trigonometric functions will be considered in some detail.

The use of an upper percentage point of the distribution of total sample size in conjunction with the expectation of the latter is proposed as a guide to the selection of the size of the initial sample when using some version of Stein's [5] two-sample procedure. It is a rapidly calculable function of the underlying population variance based on existing tables of the $\chi^2$ distribution. A rule-of-thumb is proposed to be used in making the actual selection of initial sample size. It is a simple matter to investigate the nature of the percentage point for different values of the variance over a limited range; a recommended conservative choice when the variance is not known is the selection of a large initial sample. Dantzig [2] proved the nonexistence of nontrivial tests of Student's hypothesis whose power was independent of the variance, a result extended by Stein to the general linear hypothesis. In the same paper Stein proposes a two-sample procedure whose power was independent of variance. The same two-sample method could be used to obtain a confidence interval for the mean of a normal distribution with predetermined length and confidence coefficient. Stein gave no specifications for the choice of the initial sample size, but Seelbinder [4] suggested that it be selected to minimize the expectation of the total sample. In a recent paper, Bechhofer, Dunnett and Sobel [1] used Stein's procedure for another application, noting that the variance of the total sample size increased as the size of the first sample decreased. An efficient choice of the size of the initial sample will hold the expectation of the sample small, and will further reduce the probability of an extremely large total sample. This note will explore the matter in further detail and show that an upper percentage point of the distribution of total sample size, when used inconjunction with the expectation, is a rapidly calculable guide to an efficient choice of the size of the first sample.

Let $\mathcal{F}$ be the family of all random variables on a probability space $\Omega$ taking values from a separable and complete metric space $X$. In this paper we prove that $\mathcal{F}$ is in a certain sense a closed family. More precisely, if $\{\xi_n\}$ is a sequence of $X$-valued random variables such that their probability distributions converge weakly to a probability distribution $P$ on $X$, then there exists an $X$-valued random variable on $\Omega$ with distribution $P$. An example is also given which shows that the assumption of completeness of $X$ cannot in general be dropped.