Let $\{S_n\}$ be the partial sums of a sequence (not necessarily independent) of random variables $\{X_n\}$, and let $\{m_u\}$ be a set of integer-valued random variables depending on an index $u \geqq 0$. Suppose that $m_u/u$ converges in probability to a constant as $u \rightarrow \infty$ and that $S_n$ obeys the central limit theorem (when it is normed properly, as must also be the other variables below). Anscombe [1] has shown that if the $S_n$ do not fluctuate too much, in a sense made precise below, then the random sum $S_{m_u}$ also obeys the central limit theorem. Anscombe's condition is closely related to one introduced by Prohorov [6] in connection with the Erdos-Kac-Donsker invariance principle. In Section 2 this relationship is investigated; in particular, it is shown that if the sequence $\{X_n\}$ satisfies the invariance principle then $S_{m_u}$ is asymptotically normal. The invariance principle has been proved in [2] for various dependent sequences $\{X_n\}$, to each of which this result is then applicable. In Section 3 an invariance principle is formulated and proved for the random partial sums; this result enables one to find, for example, the limiting distribution of $\max_{k \leqq m_u} S_k$. In Section 4, these theorems are applied to renewal processes.

A multivariate emigration-immigration or a Poisson-Markoff process (Bartlett [2], Ruben and Rothschild [9], Patil [8]; cf. also Bartlett [3], p. 78) is a vector stochastic process $\mathbf{n}(t) = (n_1(t), n_2(t), \cdots, n_m(t))$ in continuous time $t$ which is described by the following properties: There exists a complex of non-negative time-independent parameters, $\lambda_{rs}(r, s = 1, 2, \cdots, m, r \neq s), \lambda^\ast_r$ and $\mu_r(r = 1, 2, \cdots, m)$ such that the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) - 1, \cdots, n_s(t) + 1, n_{s + 1}(t), \cdots, n_m(t))$ in the time interval $(t, t + h)$ is $\lambda_{rs}n_r(t)h + o(h)$, for $h$ sufficiently small; the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) + 1, \cdots, n_s(t) - 1, n_{s + 1}(t), \cdots, n_m(t))$ is $\lambda_{sr}n_s(t)h + o(h)$; the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) - 1, \cdots, n_m(t))$ is $\lambda^\ast_rn_r(t)h + o(h)$; the probability of a change of $\mathbf{n}(t)$ to $(n_1(t), \cdots, n_r(t) + 1, \cdots, n_m(t))$ is $\mu_rh + o(h)$; the probability of no change is $1 - \{\sum_r \mu_r + \sum_r (\lambda^\ast_r + \sum_{s \neq r} \lambda_{rs})n_r(t)\}h + o(h).$ Finally, it is assumed that the above probabilities are independent of the past realization of the process. It is readily seen that these assumptions imply that the process is Markovian and strongly stationary. It will be convenient for our purposes to visualize the Poisson-Markoff process more concretely in the following manner: Consider a system consisting of $m + 1$ states $\{E_1, E_2, \cdots, E_m, E^\ast\}$ such that at any point in time an "individual" is in one of these $m + 1$ states. Thus, $\{E_1, \cdots, E_m\}$ may represent $m$ stages of a certain disease while $E^\ast$ represents the healthy state. A population of individuals is being studied such that the number of individuals in $E^\ast$ at any instant of time is effectively infinite. Let $\lambda_{rs}h + o(h)$ be the probability that an individual in state $E_r$ at time $t$ shall be in $E_s$ at time $t + h, \lambda^\ast_rh + o(h)$ the probability that an individual in $E_r$ at time $t$ shall be in $E^\ast$ at time $t + h$, while $\mu_rh + o(h)$ is the probability that precisely one of the (infinitely many) individuals in $E^\ast$ at time $t$ shall be in $E_r$ at time $t + h$; further, assume that the probability of more than one interchange (of an individual from any state to any other state) in $(t, t + h)$ is $o(h)$ and that there is no interaction between the individuals. Let $n_r(t)$ be the number of individuals in $E_r$ at time $t$. Then clearly the various probabilities enumerated in the first paragraph are precisely the probabilities of the various transitions of $\mathbf{n}(t)$ in the infinitesimal interval $(t, t + h)$. In this connexion, it should be noted that the emigration-immigration process is a natural generalization of the time-homogeneous birth and death process, in the sense that when $m = 1$, the former process reduces to a birth and death process, with $\lambda^\ast$ and $\mu$ representing the death and birth parameters. For $m > 1$, the $\lambda$'s and $\mu$'s may be regarded as the instantaneous mean interaction-rates between the states $E_1, \cdots, E_m, E^\ast$. Since, as remarked previously, the process $\{\mathbf{n}(t)\}$ is stationary, the expectation of $\mathbf{n}(t)$ is independent of $t$. Let, then, $\nu \equiv (\nu_1, \cdots, \nu_m) = E\mathbf{n}(t)$. Then the vector $\nu$ together with the interaction parameters ((2.1) and (2.2)) serve to specify the system completely. (Observe that $\nu$ is functionally related to the interaction parameters through (2.2).) From (2.8), $\nu$ may also be regarded as a limiting vector $(t \rightarrow \infty)$ which represents the "steady state" configuration of the system in the specific stochastic equilibrium sense implied by this equation (a configuration is defined by the vector $\mathbf{n}(t)$), while the parameters determine the approach of the system, again in a probabilistic sense, from an initial configuration to the equilibrium configuration. In Section 2, some basic properties of the Poisson-Markoff process are listed, including the mean lifetime and recurrence time of any configuration in both discrete and continuous time. Section 3 contains the main result of this paper, viz., the joint distribution of a finite set of observations of $\mathbf{n}(t)$ in discrete time from the point of view of the joint factorial moment generating function (f.m.g.f.) of this distribution. These results are of intrinsic theoretical interest and will be utilized in a subsequent paper dealing with the estimation of the fundamental interaction parameter in a singly-infinite class of emigration-immigration processes.

The problem treated is that of predicting the reliability characteristics of a complex system from data on individual components. A general model for systems maintained over a period of time is proposed, based on the idea that every system failure is induced by a component failure and corrected by the replacement of a single component. Moreover, it is assumed that components are sometimes replaced even when the system is operating correctly, in order to prevent unscheduled interruptions in operation. The assumptions which define the general model cover a number of different preventive maintenance policies, among them the following: (a) Block Changes: All components of a given type are replaced simultaneously, at times determined by a renewal process. (b) Individual Component Replacement on the Basis of Age: If a component reaches some given age without failing, it is preventively replaced. (c) System Check-Outs: If a component is used only intermittently and it fails while it is not being used, it does not induce a system failure until it is called into use. At regular intervals, those components which have failed without inducing system failure are located and replaced. (d) Marginal Testing: At regular intervals, a test is conducted to locate those components which are still operating satisfactorily but which are expected to fail in the near future. All components located by this test are replaced. It is assumed that preventive removals are regeneration points and that the performance of a component may be described by a distribution function $F(x: y)$, the probability that a component is removed by time $x$, given that it enters the system at $y$, where $x$ and $y$ are both measured from the time of the last preventive removal. $F(x: y)$ is the sum of $A(x: y)$ and $B(x: y)$, where $A(x: y)$ is the probability that the component is preventively removed by $x$ and $B(x: y)$ is the probability that the component induces a system failure by $x$. The integral equations which determine the following measures of system performance from $F(x: y), A(x: y)$, and $B(x: y)$ are developed: (1) the expected number of failures in a given time interval (2) the expected number of preventive removals in a given time interval (3) the reliability function; i.e., the probability of no failure in a given interval following a given system age. Results from Renewal Theory and the Theory of Regenerative Stochastic Processes, developed by W. L. Smith, are applied to the problem of exploring the asymptotic behavior of these quantities. Conditions sufficient for maintenance policies a, b, c, and d to meet the assumptions of the general model are precisely formulated, and the analysis necessary to derive $F(x: y), A(x: y)$, and $B(x: y)$ is carried out for each policy.

In this paper a numerical study of Chu and Hotelling's [1] method of approximating to the moments of the sample median will be made. With an introductory outline of their method in Section 2, we will proceed to apply it to various distributions, and will evaluate the degree of accuracy that can be conveniently obtained by means of it in each particular case. The numerical results will be presented in tabular form.

In this paper we obtain a recurrence relation between the pdf's (probability density functions) of order statistics. This relation is then extended to their cdf's (cumulative distribution functions) and to the expected values of a given function of each order statistic and a function of pairs of them. The interrelation between these results and other known formulas is shown. The application of the results to the preparation of tables of cdf's, expected values, variances, etc., of order statistics is indicated, and the numerical errors propagated in such computations analysed. Finally, an illustration is given. It is hoped that the unified approach presented here will provide greater insight and flexibility to procedures of numerical evaluation of expected values, etc., of order statistics.

Given $x_i, i = 1, 2, \cdots, n$, a sequence of $n$ observations, the following statistics for measuring dispersion are defined as usual: \begin{equation*} \begin{align*}s^2 &= \frac{1}{n - 1} \sum^n_{i = 1} (x_i - \bar x)^2,\qquad\big(\bar x = \sum^n_{i = 1} x_i/n\big) \\ \tag{1}\delta^2 &= \frac{1}{n - 1} \sum^{n - 1}_{i = 1} (x_i - x_{i + 1})^2,\qquad d = \frac{1}{n - 1} \sum^{n - 1}_{i = 1} |x_i - x_{i + 1}|, \\ \delta^2_2 &= \frac{1}{n - 2} \sum^{n - 2}_{i = 1} (x_i - 2x_{i + 1} + x_{i + 2}),\quad d_2 = \frac{1}{n - 2} \sum^{n - 2}_{i = 1} |x_i - 2x_{i + 1} + x_{i + 2}|.\end{align*}\end{equation*} Further, we define the following ratio criteria: \begin{equation*}\begin{align*} w^2 &= \delta^2/s^2,\qquad W = d/s,\\ \tag{2} w^2_2 &= \delta^2_2/s^2,\qquad W_2 = d_2/s, \\ u^2 &= \delta^2_2/\delta^2,\qquad U = d_2/d.\end{align*}\end{equation*} Von Neumann [12] proposed the ratio $w^2 = \delta^2/s^2$ for testing the randomness of the sequence against the alternatives of serial correlation or trend. $W = d/s$, which has the advantage of being simpler for computation, was suggested by Kamat [8] for the same purpose. Similarly, we can construct two more ratio criteria, $w^2_2 = \delta^2_2/s^2$ and $W_2 = d_2/s$, by using the second order differences. And, finally, using both first and second successive differences, it is possible to construct a third type of ratio criteria, $w^2_2 = \delta^2_2/s^2$ and $W_2 = d_2/s$, by using the second order differences. And, finally, using both first and second successive differences, it is possible to construct a third type of ratio criteria, $u^2 = \delta^2_2/\delta^2$ and $U = d_2/d$, which may also be used for detecting serial correlation or trend in successive observations of the sequence. (See Tintner [11]). Under the hypothesis of randomness, when the $x_i$ are independent normal $(\mu, \sigma)$, the distributions of the ratio statistics $w^2, w^2_2, W$ and $W_2$ are known, either in exact or approximate form (Von Neumann [12], Kamat [8] and [9].) Under the same hypothesis, Kamat [9] has shown that they are all asymptotically normal. (See also Anderson [1], Hsu [6], and Dixon [3].) The distributions, however, do not appear to have been discussed under any alternative hypotheses of non-randomness. As to comparisons of their discriminating powers, Anderson [1] has shown that, against the alternatives defined by the density function \begin{equation*}\begin{align*}p(x_1, x_2, \cdots, x_n) &= K \exp\big\lbrack -\frac{1}{2\sigma^2}\big\{(1 + \rho^2) \sum^n_{i = 1} (x_i - \mu)^2 \\ -2\rho \sum^{n - 1}_{i = 1} (x_i - \mu)(x_{i + 1} - \mu)\big\}\big\rbrack, \\ \text{or by} \\ p(x_1, x_2, \cdots, x_n) &= K \exp\big\lbrack -\frac{1}{2\sigma^2}\big\{(1 + \rho^2 - \rho)((x_1 - \mu)^2 + (x_n - \mu)^2) \\ &+ (1 + \rho^2) \sum^{n - 1}_{i = 2} (x_i - \mu)^2 - 2\rho \sum^{n - 1}_{i = 1} (x_i - \mu_(x_{i + 1} -\mu) \big\} \big\rbrack, \end{align*}\end{equation*} a statistic linearly dependent on $w^2$ is the best criterion (in the Neymann-Pearson framework) to test the hypothesis of randomness, that is to test $\rho = 0$. As mentioned by Anderson, these density functions are modifications of the density function of the circular autoregressive model, $x_i - \mu = \rho(x_{i - 1} - \mu) + e_i$, where the $e_i$ are independent normal $(0, \sigma),$ and are obtained from it by modifying the end terms in squares and products. For the autoregressive model, however, he has shown that no such best criterion exists. We are not aware of any other work on the power of these criteria. In this paper, we discuss the distributions of these ratio criteria under another alternative of serial dependence between successive observations, specified in (4) below, when the size of the sample sequence is large. It should be noted that this alternative of serial dependence is different from the alternatives considered by Anderson, and that it is not related to the autoregressive model mentioned above. First, in Section 2, we show that all these ratio criteria are asymptotically normally distributed under this alternative hypothesis. In Section 3, we obtain the means and the variances of the limiting normal distributions. In Section 4, we present numerically the power for three samples sizes, $n = 100, 200 \text{and} 400,$ and for some positive values of the serial correlation coefficient. Power curves are also exhibited for $n = 200 \text{and} 400.$ Finally, in Section 5, we compare the relative efficiencies of these ratio criteria by using the Pitman criterion of asymptotic relative efficiency, as extended by Noether [10].

For a set of variables in a given order, $s$th ante-dependence will be said to obtain if each one of the variables, given at least $s$ immediate antecedent variables in the order, is independent of all further preceding variables. If the number of variables is $p$, ante-dependence is of some order between 0 and $p - 1.$ 0th ante-dependence and $(p - 1)$st ante-dependence are equivalent to complete independence and to completely arbitrary patterns of dependence, respectively, and are defined irrespective of the ordering of the variables. 1st to $(p - 2)$nd ante-dependence are defined in terms of a specific order only. If $X_1, X_2, \cdots, X_p$ are multivariate normal, $s$th ante-dependence is equivalent to each $X_i$, given $X_{i - 1}, X_{i - 2}, \cdots, X_{i - s}, \cdots, X_{i - s - z}$, being uncorrelated with $X_{i - s - z - 1}, X_{i - s - z - 2}, \cdots, X_2, X_1$ for any non-negative $z$. In other words, the partial correlation of $X_i$ and $X_{i - s - z - k}$, given all the variables $X_{i - 1}, X_{i - 2}, \cdots, X_{i - s - z}$, is zero for all $i, k$ and $z$. The hypothesis that the covariance matrix is such that all the above partial correlations vanish will be denoted by $D_s (s = 0, 1, \cdots, p - 1)$, so that, for the multivariate normal distribution, $D_s$ denotes the hypothesis of $s$th ante-dependence. It is shown that for any set of ordered variables, normal or otherwise, $D_s$ is equivalent to the following correspondence between the regression equations of $X_i$ on all other variables, and on $X_{i - s}, X_{i - s + 1}, \cdots, X_{i = 1}, X_{i + 1}, \cdots, X_{i + s}$ only: the multiple correlations are equal, and the regression coefficients of $X_{i - s}, X_{i - s + 1}, \cdots, X_{i + s}$ are equal in both equations, all other coefficients in the former equation being zero. It is also equivalent to the $(p - s)(p - s - 1)/2$ elements in the upper right (and also lower left) corner of the inverse covariance matrix being zero. Indeed, any null hypothesis on a set of elements of the inverse covariance matrix may be formulated, and tested, as a hypothesis $D_s$ if the variables can be so ordered as to put the zero elements in the upper right and lower left corners of the inverse. Maximum likelihood estimates are derived under $D_s$ for the normal case. Likelihood ratio tests of any one $D_s$ against any other follow immediately and may be expressed in terms of the sample partial correlations. Exact distributions are not investigated, but for large samples $\chi^2$ approximations are available. Thus a sequence of tests of $D_{p - 2}$ under $D_{p - 1}, D_{p - 3}$ under $D_{p - 2}, \cdots, D_0$ under $D_1$, is obtained which, in effect, forms a breakdown of the large sample test of independence, $D_0$, under the general alternative, $D_{p - 1}$. The assumptions of ante-dependence are clearly analogous to those of Markov processes and autoregressive schemes for time series; the motivation for the study and application of these models is also similar. The present model is more general in that it relaxes the usual autoregression assumptions of equal variances and, more crucial, of equal correlations between all pairs of equidistant variables (distance being meant in terms of the order of the set or time series). This greater generality requires analysis of a sample of observations for the study of ante-dependence, whereas for autoregressive schemes there are methods of analysis based on a single observation of the time series. The ante-dependence models can be generalized to the case of several variables at each stage of the ordering. This would be analogous to the study of multiple time-series. $s$-ante-dependent sets of variables may be generated by $s$ successive summations of independent variables. This may be relevant for some applications of such models. Ante-dependence models might be applicable to observations ordered in time or otherwise. Observations on growth of organisms up to each of several ages could be analyzed in such a manner. Where growth is recorded on several dimensions, e.g., height and weight, the analysis would proceed in terms of the multidimensional generalization of the model. Other possible fields of application include batteries of psychological tests increasing in complexity, and data on the successive location of travelling objects. A study of some such applications is now under way.

The problem of classification in the case of a finite number of known distributions has been considered many times in the statistical literature, and the theory is well-developed for such problems. Some authors have considered the problem of classification in the case where some of the information about the alternative distributions has been obtained from samples. Papers concerned with this latter problem usually either present large-sample results, or else propose procedures whose use in the small-sample case is justified on intuitive or heuristic grounds (see, e.g., [4], [5], [6]). In this paper, a certain classification problem is considered in which some of the information about the alternative multivariate normal distributions has been obtained from samples. The admissibility of two "natural" decision procedures is deduced. Charles Stein has shown in [7] that "natural" procedures are not necessarily admissible when one is dealing with multivariate normal distributions. The problem is defined in Section two. In Section three, several heuristic methods of solution are considered. Each of these methods yields one or the other of two decision procedures which are called the minimum distance rule, and the restricted maximum likelihood rule, respectively. In Section four, a method for obtaining admissible translation-invariant Bayes procedures is presented. This method consists in a reparametrization, and the use of an a priori distribution of the new parameters which has a certain product measure form. In Section five, normal a priori distributions are employed to obtain a particular class of translation-invariant Bayes procedures. The results of Section five are used in Section six to show that both the minimum distance rule and the restricted maximum likelihood rule are admissible translation=invariant Bayes procedures.

The large-sample limiting shapes of the Bayes sequential testing regions of composite hypotheses are found explicityly. The result obtained is related to the Sequential Probability Ratio Test in the same way taht the likelihood ratio statistic for composite hypotheses is related to the Ne7man-Pearson test for simple hypotheses

The problem of determining a bounded length confidence interval for the zero of a regression function $R(\cdot)$ is discussed. In case $R(\cdot) = F(\cdot) - p, F$ a distribution function, $0 \geq p \geq 1,$ a closed stopping rule is given for the up-down method of experimentation. For a larger class of regression functions a closed stopping rule is given for Robbins-Monro type of experimentation. The stopping rule for the Robbins-Monro process depends on prior knowledge of an upper and a lower bound on the zero of $R(\cdot)$. It is shown that given suitable assumptions about the random variables used in experimentation finite confidence intervals for the zero of $R(\cdot)$ may be found, such confidence intervals providing an upper and a lower bound on the zero of $R(\cdot)$ with prespecified level of confidence.

(i) In the case of the choice of the largest among $k$ population means (special case $\mathscr a$ of the use of ranking methods) the assertion of Bechhofer's method ([1], [2], [3]) can be strengthened without decreasing the probability of a correct decision (Sec. 3). (ii) Bechhofer's concept of the "least favorable configuration of the population means" is studied (Sec. 4.). The result suggests at the concept is not always in accord with the underlying practical problem, but that it is in accord in the important case $\mathscr a$. (iii) an approximation is suggested for use in the case of normal populations with a common unknown variance (Sec. 5).

On the basis of a sample of observations, an investigator wants to determine the appropriate degree of a polynomial in the index, say time, to represent the regression of the observable variable. This multiple decision problem is formulated in terms used in the theory of testing hypotheses. Given the degree of polynomial regression, the probability of deciding a higher degree is specified and does not depend on what the actual polynomial is (expect its degree). Within the class of procedures satisfying these conditions and symmetry (or two-sidedness) conditions, the probabilities of correct decisions are maximized. The optimal procedure is to test in sequence whether coefficients are 0, starting with the highest (specified) degree. The procedure holds for other linear regression functions when the independent variates are ordered. The problem and its solution can be generalized to the multivariate case and to other cases with a certain structure of sufficient statistics.

The usefulness of variance component techniques is frequently limited by the occurrence of negative estimates of essentially positive parameters. This paper uses a restricted maximum likelihood principle to remove this objectionable characteristic for certain experimental models. Section 2 discusses certain necessary results from the theory of non-linear programming. Section 3 derives specific formulae for estimating the variance components of the random one-way and two-way classification models. The problem of determining the precision of instruments in the two instrument case is dealt with in section 4, and a surprising though not unreasonable answer is obtained. The remaining sections provide an algorithm for solving the problem of negative estimates of variance components for all random effects models whose expected mean square column may be thought of as forming a mathematical tree in a certain sense. The algorithm is as follows: Consider the minimum mean square in the entire array; if this mean square is the root of the tree then equate it to its expectation. If the minimum mean square is not the root then pool it with its predecessor. In either case the problem is reduced to an identical one having one less variable, and hence in a finite number of steps the process will yield estimates of the variance components. These estimates are non-negative and have a maximum likelihood property.