Given a population of $N$ units, it is required to draw a random sample of $n$ distinct units in such a way that the probability for the $i$th unit to be in the sample is proportional to its "size" $x_i$ (sampling with p.p.s. without replacement). From a number of alternatives of achieving this, one well known procedure is here selected: The $N$ units in the population are listed in a random order and their $x_i$ are cumulated and a systematic selection of $n$ elements from a "random start" is then made on the cumulation. The mathematical theory associated with this procedure, not available in the literature to date, is here provided: With the help of an asymptotic theory, compact expressions for the variance of the estimate of the population total are derived together with variance estimates. These formulas are applicable for moderate values of $N$. The reduction in variance, as compared to sampling with p.p.s. with replacement, is clearly demonstrated.

In many types of questionnaires the element of guessing is important. An approach to correcting for guessing is proposed. Here one regards the score assigned to a subject on a question as an estimate of the unknown value of the subject's knowledge on the question. This value is one when the subject knows the answer and is zero otherwise. To derive scores with minimum mean squared error, it is necessary to consider the responses of the whole population of subjects. Thus the score for a correct answer to a question depends on the proportion $p$ of correct answers in the population. In the simplest model, we assume that a subject who knows the answer, responds correctly and that others select a response at random among $r$ choices. Then an incorrect response is scored zero and a correct one is assigned a score of $\lambda/p$ where $\lambda$ is the proportion of the population who knew the answer and can be estimated using the relation $p = \lambda + (1 - \lambda)/r$. The general approach is also applied to a variety of more complicated models, each of which are examples of a specified general formulation. These models include the "pairs of questions" model, the "partial knowledge" model and the "scaled questions" model. While the method applies neatly to single questions, there are fundamental difficulties in extending it to obtain an overall or composite score for a subject on an examination consisting of many questions. This problem is discussed briefly and the simple minded procedure of totaling the scores for the individual questions is partially evaluated.

Consider a situation in which it is desired to gain knowledge about the true value of some parameter (or about the true state of the world) by means of experimentation. Let $\Omega$ denote the set of all possible values of the parameter $\theta$, and suppose that the experimenter's knowledge about the true value of $\theta$ can be expressed, at each stage of experimentation, in terms of a probability distribution $\xi$ over $\Omega$. Each distribution $\xi$ indicates a certain amount of uncertainty on the part of the experimenter about the true value of $\theta$, and it is assumed that for each $\xi$ this uncertainty can be characterized by a non-negative number. The information in an experiment is then defined as the expected difference between the uncertainty of the prior distribution over $\Omega$ and the uncertainty of the posterior distribution. In any particular situation, the selection of an appropriate uncertainty function would typically be based on the use to which the experimenter's knowledge about $\theta$ is to be put. If, for example, the actions available to the experimenter and the losses associated with these actions can be specified as in a statistical decision problem, then presumably the uncertainty function would be determined from the loss function. In Section 2 some properties of uncertainty and information functions, and their relation to statistical decision problems and loss functions, are considered. In Section 3 the sequential sampling rule whereby experiments are performed until the uncertainty is reduced to a preassigned level is studied for various uncertainty functions and experiments. This rule has been previously studied by Lindley, [8], [9], in special cases where the uncertainty function is the Shannon entropy function. In Sections 4 and 5 the problem of optimally choosing the experiments to be performed sequentially from a class of available experiments is considered when the goal is either to minimize the expected uncertainty after a fixed number of experiments or to minimize the expected number of experiments needed to reduce the uncertainty to a fixed level. Particular problems of this nature have been treated by Bradt and Karlin [6]. The recent work of Chernoff [7] and Albert [1] on the sequential design of experiments is also of interest in relation to these problems.

Linear procedures for classifying an observation as coming from one of two multivariate normal distributions are studied in the case that the two distributions differ both in mean vectors and covariance matrices. We find the class of admissible linear procedures, which is the minimal complete class of linear procedures. It is shown how to construct the linear procedure which minimizes one probability of misclassification given the other and how to obtain the minimax linear procedure; Bayes linear procedures are also discussed.

For two multivariate nonsingular normal distributions, the familiar null hypothesis of equal dispersion matrices is considered against various alternatives stated in terms of certain characteristic roots, and a physical interpretation is given for the alternatives considered. An inference procedure, which depends on similar regions and is based on one independent random sample from each of the two distributions, is proposed for the null hypothesis against each of the alternative hypotheses. Also, for three of the cases, conservative confidence bounds are obtained on one or more parametric functions which might be interpreted as measures of departure from the null hypothesis in the direction of the corresponding alternative.

A problem that seems to be of some practical interest is that of comparing $k$ experimental categories with a standard (or control) in order to determine if any of the experimental categories are better than the standard (or control) and if so, to find the best one. We might for example be concerned with evaluating several new drugs in relation to a placebo, or with evaluating several new alloys in relation to some standard alloy. This problem has been discussed in the non-sequential case in [1] and [2]. In the present paper we will be concerned with a sequential treatment of the problem. In our notation, the subscript 0 will always refer to the standard (or control), the subscript $i$ will range from 0 to $k$, and the subscript $j$ will range from 1 to $k$. Let $X_i$ be the random variable resulting from a measurement with the $i$th category $\pi_i$. We will assume the probability distribution of $X_i$ is the same for each $i$ except for a single parameter $\theta$, which might vary from category to category. The probability density of $X_i$ (or the probability of $X_i = x$ in the discrete case) will be denoted by $f(x, \theta_i)$ and this is assumed known, except for the value of $\theta_i$. For simplicity, we will suppose that the larger the value of $\theta$, the more desirable a category is. Let $R_0$ denote the relation $\theta_1 = \theta_2 = \cdots = \theta_k = \theta_0$, and let $R_j$ denote the relation $\theta_1 = \cdots = \theta_{j-1} = \theta_{j+1} = \cdots = \theta_k = \theta_0$ and $\theta_j = \theta_0 + \Delta$, where $\Delta > 0$. Let $D_0$ denote the decision that none of the experimental categories is better than the standard (or control) $\lbrack\text{that is}, \theta_s \leqq \theta_0$ for $s = 1, 2, \cdots, k\rbrack$ and let $D_j$ denote the decision that $\pi_j$ is the best experimental category and is better than the standard $\lbrack \text{that is}, \theta_j = \max (\theta_1, \theta_2, \cdots, \theta_k) > \theta_0\rbrack$. We will develop a sequential procedure for choosing one of the $k + 1$ decisions $(D_0, D_1, \cdots, D_k)$ so that the probability of selecting $D_0$ when $R_0$ is true is $\geqq 1 - \alpha$, and the probability of selecting $D_j$ when $R_j$ is true is $\geqq 1 - \beta$ for each $j, j = 1, 2, \cdots, k$. First we will treat the case where $\theta_0$ is known and $f(x, \theta)$ is arbitrary, and then consider the case where $\theta_0$ is unknown and $f(x, \theta) = (2\pi\sigma^2)^-\frac{1}{2}\exp \lbrack\frac-{1}{2}(x - \theta)^2/\sigma^2\rbrack$. A rather interesting feature of the sequential procedures that are developed is that inferior categories can be eliminated before the final stage of the experiment, which tends to decrease the number of measurements required to reach a decision. An investigation is made (when $\theta_0$ is known) of the efficiency of the sequential procedure when compared with the corresponding "best" non-sequential procedure, which indicates that a substantial saving in the average number of measurements required to reach a decision is possible with the sequential procedure.

Mixtures of distributions present two types of problems. The first is the problem of identifiability; that is, given that a distribution function $F$ is a probability mixture of distribution functions belonging to some family $\mathscr{F}$, is the mixture unique? This topic has been dealt with quite extensively in recent papers by Robbins [9], Teicher [10] and [11], and others. The second problem is that of estimating the parameters of the individual distribution functions comprising the mixture and the mixing measure. This is clearly possible only if the given mixture is identifiable. K. P. Pearson [5] and C. R. Rao [6] consider the problem of estimation for a mixture of two normal distributions and P. Rider [7 and 8] has recently constructed estimators for mixtures of two of either the exponential, Poisson, binomial, negative binomial or Weibull distributions. In each of these cases the method of maximum likelihood yields highly intractable equations. All of the above estimators have been constructed by the method of moments. In this paper moment estimators will be constructed for a mixture of two binomial distributions, $(n, p_1)$ and $(n, p_2)$. The construction presented here parallels that of Rider [8]. The limiting distributions of the estimators and their asymptotic relative efficiency will be computed. It will be shown that as the binomial parameter $n \rightarrow \infty$ the asymptotic efficiency of the moment estimators tends to unity. Finally, moment estimators will be constructed for the binomial parameters when the mixing parameter $\alpha$ is known. In this case an apparently anomalous result is obtained in that the asymptotic efficiencies of the analogous moment estimators when $\alpha$ is known tend to 0 rather than 1 as $n \rightarrow \infty$. It is pointed out that it is possible, however, to construct moment estimators whose efficiency tends to 1 in this case as well. The estimators so constructed do not depend on the known proportion $\alpha$ when $n \geqq 3$. This suggests a possible explanation of the above anomaly: This fact that the asymptotic efficiencies tend to 0 rather than 1 as $n \rightarrow \infty$ may be due to the failure of these estimators to take into account sample deviations from the true proportions in which the respective populations are present in the mixture.

Estimating the spectral density of a discrete stationary stochastic process is studied for the case when the observations consist of repeated groups of $\alpha$ equally spaced observations followed by $\beta$ missed observations, $(\alpha > \beta)$. The asymptotic variance of the estimate is derived for normally distributed variables. It is found that this variance depends not only on the value of the spectral density being estimated, but also on the spectral density at the harmonic frequencies brought in by the periodic method of sampling. Curves are presented for $\beta = 1$ showing the increase in the standard deviation and effective decrease in sample size as a function of $\alpha$.

The basic problem dealt with here is the estimation of linear regression parameters from a set of observations obscured by correlated noise. Two well-known solutions to this problem are minimum variance (or Markov, MV) and least squares (LS) estimation. Although MV is, by definition, an optimal method, LS possesses two distinct advantages which cause it to be used more frequently in practice: (1) computational simplicity and (2) the fact that it does not require knowledge of the correlation matrix of the noise, which in many cases is actually unknown. Therefore, a comparative study of these two methods to determine how much is lost by use of LS instead of MV is of value. In this connection, Grenander and Rosenblatt [1] have derived important asymptotic properties of LS and MV estimates when the noise is a stationary random process. The approach here is somewhat different from theirs, and no assumption regarding stationarity is made. The essence of this analysis is to re-formulate LS and MV in terms of the spectrum of the noise correlation matrix. This procedure offers some new insights into the nature of LS and MV and the differences between them. For example, it shows when they yield the same result and when LS performs worst compared with MV. It also exhibits the roles played by the maximum and minimum eigenvalues of the noise correlation matrix in setting bounds on the covariance matrices of both LS and MV estimates.

The properties of the power function of the test of tendency or the "quadrant measure of association" are studied. A formula giving a lower bound for the exact power function of this test with respect to normal bivariate one-tailed $(\rho > 0)$ alternatives is obtained. Further, an approximate formula for this "minimum power" is suggested. Some numerical values are calculated from the exact and approximate formulae for this lower bound of the power function. A comparison with Konijn's approximation is presented.

It is now coming to be generally agreed that in testing for shift in the two-sample problem, certain tests based on ranks have considerable advantage over the classical $t$-test. From the beginning, rank tests were recognized to have one important advantage: their significance levels are exact under the sole assumption that the samples are randomly drawn (or that the assignment of treatments to subjects is performed at random), whereas the $t$-test in effect is exact only when we are dealing with random samples from normal distributions. On the other hand, it was felt that this advantage had to be balanced against the various optimum properties possessed by the $t$-test under the assumption of normality. It is now being recognized that these optimum properties are somewhat illusory and that, under realistic assumptions about extreme observations or gross errors, the $t$-test in practice may well be less efficient than such rank tests as the Wilcoxon or normal scores test [6], [7]. Rank tests were naturally developed first for the simple two-sample problem, but in practice experiments for the evaluation of population or treatment differences are seldom of this form. To secure the advantages of increased homogeneity and the resultant increased precision, it is more customary to stratify the populations or to divide the experimental subjects into blocks, using a (generalized) randomized block design. In other experiments where additivity of certain effects is assumed, a design of the Latin square type may be appropriate. It is the purpose of the present paper to provide a method for constructing rank tests for such designs. The basic idea of the method is described in Section 3. The main body of the paper is concerned with the application of the method to the comparison of two treatments using a Wilcoxon-type test statistic. The exact distribution of this statistic is discussed in Section 4, and its asymptotic distribution in Sections 5 and 6. Some remarks concerning the efficiency of the test are given in Section 7. Finally, Section 8 illustrates the application of the method to the comparison of more than two treatments, for such designs as incomplete blocks and Latin squares.

This paper is concerned with two sample rank tests for scale alternatives. The two samples are assumed to have continuous distribution functions with the difference in respective location parameters (medians) known. Various rank tests are considered and compared from the point of view of limiting Pitman efficiency for normal and nonnormal alternatives. Among the tests considered is a test with efficiency one relative to the $F$-test for normal alternatives. Tables are given to facilitate its use. Small sample power and efficiency for normal alternatives are computed for equal sample sizes of 5. The small sample efficiency values differ appreciably from the limiting value; this deficiency of power appears to derive from the use of ranks per se rather than from the use of a rank test that is not optimal among rank tests. Lastly, a rank test is proposed for particular alternatives which is most powerful for rectangular densities. It is a simple test which is seen to have surprisingly good power for normal alternatives.

The paradox of almost certain rejection of the null hypothesis in the Chi Square test-of-fit, when many observations are used, has been pointed out by Cochran [4], and largely removed by Lehmann and Hodges [7]. The same paradox arises in most tests of goodness-of-fit. In this paper the Kolmogorov-Smirnov tests are modified to remove this difficulty and some properties of this modification are investigated. In particular, a rigorous method for choosing sample size (Theorem 3.2 and corollaries) is presented. Given independent random variables $X_1, \cdots, X_n$ with common distribution function $F$, suppose that we desire to determine whether or not $F$ is in some class $\mathscr{H}_0$. If we are only interested in whether $F$ is close, in some sense, to some distribution function in $\mathscr{H}_0$, we can let $\mathscr{H}^\ast_0 \supset \mathscr{H}_0$, where $\mathscr{H}^\ast_0$ is the class of distribution functions "close" to those in $\mathscr{H}_0$, and test the more reasonable hypothesis that $F \epsilon \mathscr{H}^\ast_0$. In what follows we consider tests based on the uniform metric $d_1$, given by $d_1(F, H) = \sup_x |F(x) - H(x)|$, where $\mathscr{H}_0$ consists of a single distribution function $F_0$.

The statistic described in the title may be used to provide a test of the hypothesis that a population has a prescribed continuous distribution function. It differs from that proposed by Kuiper for use with distributions on a circle only in that the usual empirical distribution function is replaced by Pyke's modification. This change and a theorem of Sparre Andersen make possible a computation of the exact distribution for finite sample size along the lines of the computation of the distribution of Kolmogorov's statistic. A brief comparison is made in Section 1 of this statistic, Kolmogorov's, and a statistic studied by Sherman, as distances between hypothetical and empirical distribution functions. The asymptotic distribution, due to Darling and Kuiper, appears in Section 3. Tables are included of the distribution for sample sizes 1 through 20.

As has been noted by several authors, when a multivariate normal distribution with correlation matrix $\{\rho_{ij}\}$ has a correlation structure of the form $\rho_{ij} = \alpha_i \alpha_j (i \neq j)$, where $-1 \leqq \alpha_i \leqq + 1$, its c.d.f. can be expressed as a single integral having a product of univariate normal c.d.f.'s in the integrand. The advantage of such a single integral representation is that it is easy to evaluate numerically. In this paper it is noted that the $n$-variate normal c.d.f. with correlation matrix $\{\rho_{ij}\}$ can always be written as a single integral in two ways, with an $n$-variate normal c.d.f. in the integrand and the integration extending over a doubly-infinite range, and with an $(n - 1)$-variate normal c.d.f. in the integrand and the integration extending over a singly-infinite range. We shall show that, for certain correlation structures, the multivariate normal c.d.f. in the integrand factorizes into a product of lower-order normal c.d.f.'s. The results may be useful in instances where these lower-order integrals are tabulated or can be evaluated. One important special case is $\rho_{ij} = \alpha_i \alpha_j$, previously mentioned. Another is $\rho_{ij} = \gamma_i/\gamma_j (i < j)$, where $|\gamma_i| < |\gamma_j|$ for $i < j$. Some applications of these two special cases are given.

Renyi [19] gives a set of seven postulates which a measure of dependence for a pair of random variables should satisfy. Of the dependence measures considered by Renyi, only Gebelein's [5] maximal correlation, $S_P$, satisfies all seven postulates. Kramer [10] in considering the uncertainty principle in Fourier analysis [11] generalizes the Gebelein maximal correlation to the case of arbitrary pairs of $\sigma$-algebras; and asks whether this generalization is equivalent to Shannon's mutual information, $C_P$, [4, 9, 21] for pairs of $\sigma$-algebras--equivalent in the sense of preserving order. The object of this note is to compare $S_P$ and the two normalizations $C'_P$ and $C"_P$, of $C_P$, as dependence measures for strictly positive probability spaces (which are necessarily generated by random variables). It is found that for such spaces with the proper finiteness restrictions (a) (Thm 5.1) $0 \leqq S_P, C'_P, C"_P \leqq 1$; (b) (Thm 5.2) $S_P = 0 \text{iff} C'_P = 0 \text{iff} C"_P = 0$ iff the random variables are independent; (c) (Thm 5.4) $S_P = 1$ if the two generated algebras have a nontrivial intersection (the conditions are equivalent for finite algebras); $C'_P = 1$ iff one of the random variables is a function of the other; and $C"_P = 1$ iff the random variables are functions of each other; and, consequently, (d) (Thm 5.5) there exist probability spaces for which the dependence measures are not equivalent. The paper is divided into six sections. Section 1 contains the introduction and summary. Section 2 introduces the terminology, notation and preliminaries. Section 3 treats $S_P$ and the Renyi postulates. In Section 4, the basic Shannon-Feinstein-Khinchin mutual information is extended to strictly positive measure spaces, not necessarily finite. The comparison of the dependence measures and postulate modifications are given in Section 5. Finally, in Section 6 some extensions and open problems are mentioned.

In [5], the concept of statistical sufficiency is studied within a general probability setting. The study is continued here. The notation and definitions of [5] are used. Here we give an example of sufficient statistics $t_1$ and $t_2$ such that the pair $(t_1, t_2)$ is not sufficient. The example also has the property that, in a sense to be made precise, no smallest sufficient statistic containing $t_1$ and $t_2$ exists. In Example 4 of [5], sufficient subfields $\mathbf{A}_1$ and $\mathbf{A}_2$ are exhibited such that $\mathbf{A}_1 \vee \mathbf{A}_2$, the smallest subfield containing $\mathbf{A}_1 and \mathbf{A}_2$, is not sufficient. Such an example is given here with the even stronger property that no smallest sufficient subfield containing $\mathbf{A}_1$ and $\mathbf{A}_2$ exists. Let $(X, \mathbf{A}, P)$ be the probability structure under consideration. Here $X$ is a set, $\mathbf{A}$ is a $\sigma$-field of subsets of $X$, and $P$ is a family of probability measures $p$ on $\mathbf{A}$. Let $\mathbf{N}$ be the smallest $\sigma$-field containing the $P$-null sets and let $K$ be the collection of sufficient subfields of $A$ containing $N$. (Restricting attention to sufficient subfields containing $N$ is technically convenient. Note that any sufficient subfield is equivalent, in the usual sense, to one containing $N$.) Some of the properties of $K$ can be described in the language of lattice theory as follows. Let $L$ be the set of subfields (= sub-$\sigma$-fields) of $\mathbf{A}$. Then $L$, partially ordered by inclusion, is a complete lattice. (Our terminology is essentially that of Birkhoff [4].) Example 4 of [5], mentioned above, shows that $K$ is not always a sublattice of $L$. The example given below shows more: The set $K$, partially ordered by inclusion, is not always a lattice in its own right. Note, however, that if $H$ is a finite, or even countable, subset of $K$, then the greatest lower bound of $H$ relative to $L$ exists and is in $K$ ([5], Corollary 2). The difficulty is with the least upper bound. There is less difficulty if $\mathbf{A}$ is separable. Corollaries 2 and 4 of [5] indicate that if $A$ is separable, then $K$ is a $\sigma$-complete sublattice of $L$. This is about as strong a result as could be expected here. For even if $\mathbf{A}$ is separable, $\mathbf{K}$ is sometimes neither complete nor conditionally complete: Each of the nonsufficient subfields exhibited in Example 1 of [5] is easily seen to be both the greatest lower bound of a subset of $K$ and the least upper bound of a subset of $K$. There is no difficulty if $P$ is dominated. If $P$ is dominated, then $K$ is a complete sublattice of $L$. This follows easily from the existence in this case (Bahadur [2], Theorems 6.2 and 6.4; Loeve [6], Section 24.4) of a subfield $\mathbf{A}_0$ in $K$ such that $K = \{\mathbf{B} \mid \mathbf{B} \epsilon L, \mathbf{A}_0 \subset \mathbf{B}\}$.

This paper introduces a special calculus for the analysis of factorial experiments. The calculus applies to the general case of asymmetric factorial experiments and is not restricted to symmetric factorials as is the current theory which relies on the theory of finite projective geometry. The concise notation and operations of this calculus point up the relationship of treatment combinations to interactions and the effect of patterns of arrangements on the distribution of relevant quantities. One aim is to carry out complex manipulations and operations with relative ease. The calculus enables many large order arithmetic operations, necessary for analyzing factorial designs, to be partly carried out by logical operations. This should be of importance in programming the analysis of factorial designs on high speed computers. The principal new results of this paper, aside from the new notation and operations, are (i) the further development of a theory of confounding for asymmetrical factorials (Section 4) and (ii) a new approach to the calculation of polynomial regression (Section 5). In particular, the use of the calculus enables one to write the inverse matrix of the normal equations for a polynomial model as a partitioned matrix. As a result it only requires inverting matrices of smaller order.

This paper deals with the analysis and constructions of Symmetrical Unequal Block (SUB) arrangements with two unequal block sizes. The analysis of these designs is obtained on the assumption of equal intrablock error variances for blocks of different sizes. Various methods of constructing these arrangements from known incomplete block designs are considered in this paper. The method of constructing SUB arrangements by the method of finite differences is also considered in this paper.

The study of laws of random variables is facilitated by various methods of representing these laws. The distribution function and characteristic function have played an important role, and the functional representation of the law of a random variable $X$ as a mapping $T$ of the bounded Borel functions into the real line given by $Tg = Eg(X)$ connects many problems in probability theory with functional analysis. Similarly, in the study of conditional probability laws various representations of these conditional laws are desirable. In particular, in the study of Markov processes an operator representation has proven most useful. In this paper we develop the representation of the conditional law of a random variable in a way analogous to the functional representation mentioned above. Thus, if $X$ is a random variable on a probability space $(\Omega, \mathcal{A}, P)$ and $\mathcal{A}^\ast$ is a sub-$\sigma$-field of $\mathcal{A}$, then we introduce the $\mathcal{A}^\ast$ conditional operator $T$ of $X$ mapping the bounded Borel functions into $L_\infty(\Omega, \mathcal{A}^\ast, P)$ and given by $T_g = E^{\mathcal{A}\ast} g(X)$. The first four sections of the paper develop a rudimentary theory of such operators. The use of these operators in probability theory leads one to consider various operator topologies, etc., weaker than those usually studied in functional analysis. Most of the material presented in this development is quite elementary, and many properties of these operators having potential interest have not even been mentioned. Nevertheless, it is hoped that this exposition may suggest further use of the operator representation of conditional distributions in probability theory, and that the properties of these operators relevant to probability theory will be investigated more systematically. In the final section conditional probability operators are applied to a mixing problem. A stationary process is said to have central structure if, under conditions similar to those of the central limit problem for independent random variables, conditional limit laws given the invariant $\sigma$-field are infinitely divisible. It is shown that central structure is closely related to a type of uniform ergodicity.

In this paper the relation between weak convergence of a sequence of measures and uniform convergence over certain classes of continuity sets (or uniform convergence of the integrals over certain classes of continuous functions) is studied. These results are applied to obtain laws of large numbers for random functions and generalizations of the Glivenko-Cantelli lemma.

Consider an initial real interval $\lbrack 0, x\rbrack, x > 0$, and place in it a random subinterval $I(x)$ defined by a pair of random variables $(U_x, V_x)$; the former being the length of $I(x)$ and the latter its lower boundary point. The set $\lbrack 0, x\rbrack - I(x)$ consists of two intervals of lengths $x_1$ and $x_2$, in which there are in turn placed random subintervals $I(x_1)$ and $I(x_2)$ defined by pairs of random variables $(U_{x_1}, V_{x_1})$ and $(U_{x_2}, V_{x_2})$. The process of placing random subintervals in $\lbrack 0, x\rbrack$ is thus continued. Under the assumptions that the subintervals cannot overlap, and that their lengths are uniformly bounded away from zero, the procedure must terminate after a finite number of steps. Let $N(b, x)$ denote the number of subintervals of $\lbrack 0, x\rbrack$ of length at least $b$, in the terminal state. The asymptotic behaviour of the moments of $N(b, x)$ is here studied as $x \rightarrow \infty$. It is shown that under fairly general conditions the mean approaches a linear function of $x$ at the rate $x^{-n}$, for any integer $n > 0$. Under the further condition that $V_x$ is a family of uniform distributions the exact form of the linear relation is determined. In the last section it is indicated how this result can be extended to some more general distributions. A similar but less precise result is proved for the higher moments, the convergence rate $x^{-n}$ not being established for this case.

We consider a system with a finite number $S$ of states $s$, labeled by the integers $1, 2, \cdots, S$. Periodically, say once a day, we observe the current state of the system, and then choose an action $a$ from a finite set $A$ of possible actions. As a joint result of the current state $s$ and the chosen action $a$, two things happen: (1) we receive an immediate income $i(s, a)$ and (2) the system moves to a new state $s'$ with the probability of a particular new state $s'$ given by a function $q = q(s' \mid s, a)$. Finally there is specified a discount factor $\beta, 0 \leqq \beta < 1$, so that the value of unit income $n$ days in the future is $\beta^n$. Our problem is to choose a policy which maximizes our total expected income. This problem, which is an interesting special case of the general dynamic programming problem, has been solved by Howard in his excellent book [3]. The case $\beta = 1$, also studied by Howard, is substantially more difficult. We shall obtain in this case results slightly beyond those of Howard, though still not complete. Our method, which treats $\beta = 1$ as a limiting case of $\beta < 1$, seems rather simpler than Howard's.

Let $\{Z(t), t > 0\}$ be a separable, continuous time Markov Process with stationary transition probabilities $P_{ij}(t), i, j = 1, 2, \cdots, M$. Under suitable regularity conditions, the matrix of transition probabilities, $P(t)$, can be expressed in the form $P(t) = \exp tQ$, where $Q$ is an $M \times M$ matrix and is called the "infinitesimal generator" for the process. In this paper, a density on the space of sample functions over $[0, t)$ is constructed. This density depends upon $Q$. If $Q$ is unknown, the maximum likelihood estimate $\hat{Q}(k, t) = \|\hat{q}_{ij}(k, t)\|$, based upon $k$ independent realizations of the process over $\lbrack 0, t)$ can be derived. If each state has positive probability of being occupied during $\lbrack 0, t)$ and if the number of independent observations, $k$, grows larger ($t$ held fixed), then $\hat{q}_{ij}$ is strongly consistent and the joint distribution of the set $\{k^{\frac{1}{2}}(\hat{q}_{ij} - q_{ij})\}_{i \neq j}$ (suitably normalized), is asymptotically normal with zero mean and covariance equal to the identity matrix. If $k$ is held fixed (at one, say) and if $t$ grows large, then $\hat{q}_{ij}$ is again strongly consistent and the joint distribution of the set $\{t^{\frac{1}{2}}(\hat{q}_{ij} - q_{ij})\}_{i \neq j}$ (suitably normalized), is asymptotically normal with zero mean and covariance equal to the identity matrix, provided that the process $\{Z(t), t > 0\}$ is positively regular. The asymptotic variances of the $\hat{q}_{ij}$ are computed in both cases.

In [3], a general model for the reliability analysis of systems under various preventive maintenance policies is postulated and analyzed. The integral equations that determine the expected number of failures, the expected number of preventive removals, and the survival probability function are developed. In the present paper, a particular model of a system subject to marginal testing is considered and explicit values of these performance measures are obtained. Under a marginal testing policy, the system is maintained in operating condition by replacing all failed components as soon as they fail and, at regular intervals, conducting a test 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. In this model, it is assumed that a component may be in any one of $n + 1$ states, $0, 1, \cdots, n$, and, during normal operation, these states constitute a continuous-parameter Markov process in which state $n$ is the failed state. When a component enters state $n$, it is immediately replaced by one in state 0. The marginal test detects the state and states $k, k + 1, \cdots, n - 1$ are considered marginal. The test is performed at fixed intervals, and, if a component is found in the marginal state, it is replaced by one in state 0. Since this model provides for transitions from any operative state to any other state, recovery from the marginal state to the good state is permitted, a characteristic which was not allowed in the model of [3]. In addition, a choice of the level at which the component is considered marginal is permitted. The loss of generality lies in the assumption that the process is Markovian. As in [3], it is assumed that there is no dependence between transitions in different component positions and that every system failure is corrected by the replacement of one component, so that the problem of determining system performance measures is reduced to the problem of determining the corresponding quantities for a single component position. In the analysis of this model, we shall first analyze the Markov process in the absence of marginal tests and determine the matrix of probabilities $H_{ij}(t)$ that a component is in state $j$ at $t$ given that it is in state $i$ at 0, the distribution function $F_i(t)$ of time to failure for a component initially in state $i$, and $\mathscr{B}_i(t)$, the expected number of failures in a component position by $t$, given that the component is in state $i$ at 0. It is then seen that, with the marginal test performed at intervals of length $T$ and replacement of marginal components, the states of the component just before successive test points constitute a discrete-parameter Markov chain. The matrix of transition probabilities $p_{ij}$ for this chain and the probability $P_i(r)$ that the component is found in state $i$ at the $r$th test point are determined. Then the expected number of preventive removals in a component position, $U_p(t)$, may be expressed as the sum of the probabilities of finding the component marginal over all test points before $t$ while the expected number of failures, $U_f(t)$, is expressible in terms of the $P_r's$ and the $\mathscr{B}_i's$. The asymptotic values of $U_p(t)$ and $U_f(t)$ are determined by the stationary properties of Markov chains. Finally, the Markovian nature of this model makes it relatively easy to express $R(t; x)$, the probability of no failure in a component position in an interval $t$ following system age $x$, in terms of the matrices discussed previously.

Two minimax "regret" criteria for global optimality of designs in terms of estimation of the entire regression function are shown to be equivalent to minimizing the generalized variance. (The equivalence for the minimax criterion without modification was proved by Kiefer and Wolfowitz [9].) A consequent algorithm which is helpful in computing optimum designs is given.

M. C. K. Tweedie [2] defined the inverse Gaussian distributions via the density functions \begin{equation*}\tag{1}f(x; m, \lambda)= \lbrack\lambda/(2\pi x^3)\rbrack^{\frac{1}{2}} \exp \lbrack -\lambda(x - m)^2/(2m^2x)\rbrack\quad\text{for}\quad x > 0 = 0\quad\text{for} x \leqq 0\end{equation*} The parameters $\lambda$ and $m$ are positive. The corresponding densities reflected about the origin, and with $\lambda$ and $m$ negative, may also be considered as in the Inverse Gaussian family. The characteristic function of the Inverse Gaussian distribution with parameters $\lambda, m$ is \begin{equation*}\tag{2}\phi(t) = \exp \lbrack\lambda\{1 - (1 - 2im^2t\lambda^{-1})^{\frac{1}{2}}\}/m\rbrack,\quad i = \sqrt{-1}\end{equation*} for all real values of $t$. If $x_1, x_2, \cdots, x_n$ are $n$ independent observations from (1), then $y = \sum^n_{j=1} x_j$ and $z = \sum^n_{j=1} x^{-1}_j - n^2y^{-1}$ are independently distributed. The distribution of $y$ is $f(y, nm, n^2\lambda)$ and that of $\lambda z$ is Chi-Square with $(n - 1)$ degrees of freedom. In this note, we prove that, if $x_1, x_2, \cdots, x_n$ are independently and identically distributed variates, with the existence of certain moments (different from zero), and if $y$ and $z$ are independently distributed, then the distribution of $x_j$ is Inverse Gaussian.

This note presents a simplified expression for the exact null distribution of the Hodges bivariate sign test. The form is suitable for computing both for small and large sample size and gives the limiting distribution easily. A small table is given for the limiting distribution. The Bahadur limiting efficiency of the test is computed relative to the bivariate Hotelling $T^2$ test for normal alternatives. The value $2/\pi$ is obtained, which is the same as for the one dimensional sign test relative to the $t$-test.

Let $X_1, X_2, \cdots, X_n$ be $m$-dimensional statistically independent random vectors with common distribution function $F$. It is frequently desirable to test the hypothesis that $F$ is a member of some class of distribution functions $\mathscr{H}_0$. For the scalar case, $(m = 1)$, much research has been done; see for example [1], [2], [3]. For $m > 1$ comparatively little has been accomplished, and a useful extension of the techniques used for $m = 1$ awaits the solution of certain problems in stochastic processes with a vector parameter; see, for example, [4]. In this paper consistent tests are developed for any given class $\mathscr{H}_0$. These tests can be constructed to have size $\alpha$ and prescribed power $1 - \beta$ against alternatives whose probability assignment to at least one of a certain given class of sets $\{B(v)\}$ differs from that of each member of $\mathscr{H}_0$ by at least a prescribed value $K$. The range of such alternatives is seen in Section 3 to be rather wide, so that at least in theory, the suggested tests would seem to be rather useful. The tests are constructed by mapping the set of all $m$-dimensional distribution functions in a one to one measurable manner into a subset of the set of one-dimensional distribution functions. Such mappings are, of course, reasonably well known; see Halmos [7], p. 153. The purpose of this note is to show how such mappings offer sufficient flexibility for the construction of a class of tests which are useful for most ordinary purposes. Simpson [6] suggested tests based on mapping bivariate distributions into univariate distributions. However no mention was made there of consistency, or of power in terms of the type of alternative here mentioned.