One of the unpleasant facts about statistical decision problems is that they are generally much too big or too difficult to admit of practical solutions, a fact that is threatening to widen even further the gap between the theory and application of this brave new discipline. Briefly, the situation is this. For each possible decision procedure $\varphi$, the statistician is concerned only with the values $\rho (\omega, \varphi)$ of the risk function as $\omega$ ranges over the set $\Omega$ of all possible states of nature, so that a choice of a decision procedure amounts to a choice of a risk function. The obvious difficulty of comparing functions in the search for a best procedure now arises, constituting a major problem for the statistician. The Bayes and minimax principles, it should be noted, represent but two extremes to which the statistician can go to get around this difficulty rather than to meet it, the one assuming complete knowledge of an a priori probability distribution $\lambda$ of the possible states $\omega$, the other assuming least favorable circumstances about $\omega$, so that in either case one considers only a single number per procedure rather than the entire function--the Bayesian the average risk with respect to $\lambda$, the minimax-man in all timidity the supremum of the risk function--comparisons thus becoming trivial in principle and obliging one to look simply for that procedure which minimizes these numbers. Inasmuch as the situations occurring in practice with regard to prior knowledge about $\omega$ usually lie between the two extremes just described, to this extent at least both principles are open to criticism. In view of the nature of the difficulty in choosing among procedures, the notion of admissible or complete classes of strategies is generally felt to provide the most satisfactory solution. Whereas it may be difficult to say what to do in a statistical decision problem, it is generally easier to say what not to do, so that the statistician separates out from consideration all inadmissible strategies and presents the practical man with what is at best a minimal essentially complete class of procedures. The choice of one from among these admissible procedures is then left to the best judgment, intuition, and past experience of the practical man. If the class is a small one, we have then achieved everything one can ask for, for the actual choice will then easily be made. The difficulty, however, is that the classes are usually much too big to be of real help. The trouble, in a word, lies in the fact that it is the space $\Omega$ itself of possible states that is generally too big: one simply cannot look at and assess all the values of $\omega$ for each decision function $\varphi$. Now when problems turn out to be too big for practical purposes, it is natural to look for ways of cutting them down to size by methods of simplification or approximation in which very little of the original problem is lost. It is precisely such a cutting down or slicing up of the problem that we propose to treat in this paper, in the hope that it may help bring the theory and practice of statistical decision functions somewhat closer together. The Modified Minimax Principle. The minimax principle, which looks only at the single value $\sup_{\omega\rho}(\omega, \varphi)$, suffers from the defect of being an over-simplification. Yet it suggests, by means of a simple modification, a natural way of approximating to the problem. This is to cut $\Omega$ up, to partition it into sets or "slices" $\Omega_s, s$ running over an index set $S$, and then look at $\sup_{\omega\varepsilon\Omega_s\rho}(\omega, \varphi) = \alpha(s, \varphi)$ for each $s$ in $S$, so that corresponding to the partition $\Omega = \cup_{s\varepsilon S}\Omega_s$ we look only at the values $\alpha(s, \varphi)$ for $s \varepsilon S$ instead of at $\rho(\omega, \varphi)$ for all $\omega$. The range of $\rho_\varphi$ is thus replaced by the smaller range of $\alpha_\varphi$, making comparisons of procedures that much easier. It is this slicing up of $\Omega$ (or its replacement by the smaller class $S$) and consequent simplification of the risk functions in the above way that we call the modified minimax principle. (By way of analogy, one might think of upper Darboux sums approximating Riemann integrals.) The reduced game can then be treated as any other game: one can play Bayes or minimax in it, or attempt to delineate its admissible strategies. If the "slicing principle" used is such that the supremum of $\rho_\varphi$ over each slice is not much different from the values within the slice, or has some other reasonable property, then very little is lost. The question of what are reasonable or natural slicing principles is clearly of primary importance here, and we shall present what we believe are several of them. The theory of invariance provides us with the most powerful of these slicing principles and will play a central role in our considerations. The natural slicing of $\Omega$ into its orbits under the group leads us to what appears to be the best possible generalization of the Hunt-Stein theorem, and to its most natural setting: namely to the theorem that under certain regularity conditions which are often met in practice the invariant procedures form a complete class in the sense of the sliced up risk functions. The theory of composite hypotheses is also discussed in this light, and an example of theoretical interest is given illustrating these concepts, in which a difficult problem undergoes a striking simplification. Finally, the use of previous experience as a slicing principle is discussed, and related to a purely game-theoretic model which we have constructed for the modified minimax principle and which we have called a mixed game. Though these slicing principles appear to be among the most important, the search for others continues. A method of approximation and simplification having been established, it remains to be seen whether these principles and available numerical methods can be combined to make an effective instrument in practice.

The use of incomplete block designs for estimating and judging the significance of the difference of treatment effects is now standard. Fisher and Yates [2] have provided a complete table of balanced incomplete block designs (BIB) for the value of $r, k \leqq 10$. In this paper a method of constructing a special class of partially balanced incomplete block designs (PBIB) from the known BIB designs is given. Vartak [6] and Sillito [4] have used the Kronecker product of matrices to construct statistical designs. Their method and the method given in this paper differ only in the fact that in their method two distinct elements of a matrix are replaced by two distinct matrices, while in the method considered in this paper, two or more distinct elements of a matrix are replaced by different matrices. All the PBIB designs considered in this paper are with three associate classes, and the rectangular association scheme for $\nu = pq$ treatments are written in $p$ rows and $q$ columns so that treatments in the same row are the first associates, treatments in the same column are the second associates and the rest are the third associates.

A normal regression experiment can be represented by \begin{equation*}\tag{1.1} Y_i = \sum_{j=1}^k X_{ij} \theta_j + \eta_i\qquad (i = 1, \cdots, n)\end{equation*} where $\{\eta_i/i = 1, \cdots, n\}$ is a set of normally distributed random variables with zero means and non-singular dispersion matrix $C, \theta = (\theta_1, \cdots, \theta_k)$ is the parameter-vector of interest and $X = (X_{ij})$ is a known $n \times k$ matrix which will be called the allocation matrix. The rows of $X$ will be called the allocation vectors. We denote the experiment by $\varepsilon(X, C)$. We assume that $C$ is known; generally it will be a function of $X, C(X)$. The particular realisation of $Y$ will be denoted $y$. The matrix $F = X'C^{-1}X$ is the Fisher-information-matrix of $\varepsilon(X, C)$. When $F$ is non-singular, one answer to the question "What information does $y$ give about $\theta$?" is to quote $F^{-1}$, the dispersion matrix of the maximum-likelihood-estimates of $\theta$. A strong argument in favour of this is that $F^{-1}$ is independent of both $\theta$ and $y$. The fact that it is independent of $\theta$ means that the answer is not "local"; the fact that it is independent of $y$ leads to simplicity. This approach is taken by Box and Hunter [1] in their work on rotatable designs. However, we must accept the fact that many experimenters wish to have a one-dimensional answer to the question i.e. we must associate with $\varepsilon(X, C)$ a single number which we call the "information". For instance Elfving [5] has developed the use of trace $F^{-1}$. In this paper we adopt the measure of information introduced by Lindley [7]. In Section 2 we generalise Lindley's treatment of the regression situation to include the singular case, explain the uses of the measure and compare it with that of Elfving. Section 3 deals with the analogue of Elfving's main theorem. Theorems 4.1 and 4.2 of Section 4 provide links with the traditional variance approach. In Section 5 we derive the asymptotic form of the measure as the $n$ of (1.1) increases and show that this form can be derived also from Neyman-Pearsonian theory. In Section 6 the influence of nuisance parameters is discussed and an analogue of a theorem of Chernoff [2] is established.

This note is concerned with estimation of the Shannon-Wiener measure of information. Low bias estimates are obtained, and their bias and variance. These estimates are extended to the case where the number of possible values of the random variable is not known. The estimates are compared asymptotically with the maximum likelihood estimates. They are also compared with the minimax estimates (for squared error loss function) for a few special cases where these are easily found.

The subject of minimum variance unbiased estimation has received a great deal of attention in the statistical literature, e.g., in the papers of Bahadur [2], Barankin [3], and Stein [14]. The emphasis in these papers has typically been placed on the existence and construction of minimum variance unbiased estimators when the sampling plan to be used was given in advance. In this paper, criteria are developed for the selection of an appropriate sampling plan for the family of binomial distributions. Thus, independent observations are to be taken on the random variable $U$ so distributed that \begin{equation*}\tag{1.1} \operatorname{Pr} \{U = 1 | p\} = p,\quad \operatorname{Pr} \{U = 0 | p\} = 1 - p = q,\end{equation*} where $p$ lies in the open interval $0 < p < 1$, and the value of a given function, $g(p)$, is to be estimated. The problem considered here is that of determining a sampling plan and an unbiased estimator of $g(p)$ that are optimal, in some sense, at a specified value, $p_0$, of $p$. Optimality will depend, not only on the variance of the estimator, but also on the average sample size of the plan. A sampling plan, $S$, and estimator, $f$, will be considered optimal at $p_0$ if, among all procedures with average sample size at $p_0$ no larger than that of $S$, there does not exist an unbiased estimator with smaller variance at $p_0$ than that of $f$. The basic tool to be used is the information inequality (see Lemma 2.7 and the discussion following it) which provides a lower bound for the variance of an estimator in terms of its expected value and the average sample size of the sampling plan. If, at $p_0$, this lower bound is attained for a particular estimator and sampling plan, it may be immediately concluded that they are optimal at $p_0$. Such an estimator is said to be efficient at $p_0$. In Section 2, various definitions, assumptions, and fundamental facts to be used throughout the paper are collected. In Section 3 it is shown that the single sample plans and the inverse binomial sampling plans are the only ones that admit an estimator that is efficient at all values of $p$. In Section 4 some techniques are given that are often useful in the analysis of inverse binomial sampling plans. In Section 5 relationships between the average sample size of a sampling plan and the functions of $p$ that are estimable optimally are explored. In Sections 6 and 7 it is shown that there can exist two distinct sampling plans with the same average sample size for all $p$ and some comparisons are made of such plans. In Section 8 a new characterization of completeness is given for bounded sampling plans and it is shown that the dimension of the linear space of unbiased estimators of 0 can be determined simply by counting the number of boundary points. It is further shown that for a wide class of plans, the estimators that are efficient at a given value of $p$ do not have uniformly minimum variance, although non-trivial uniformly minimum variance estimators do exist. After this paper had been accepted for publication I learned that R. B. Dawson had obtained expression (4.13) and Theorems 8.2 and 8.4, as well as various other interesting results related to the material of Section 8, in his Ph.D. thesis, "Unbiased tests, unbiased estimators, and randomized similar regions," Harvard University, May, 1953.

The problem of selecting the multinomial event which has the highest probability is formulated as a multiple-decision selection problem. Before experimentation starts the experimenter must specify two constants $(\theta^{\ast}, P^{\ast})$ which are incorporated into the requirement: "The probability of a correct selection is to be equal to or greater than $P^{\ast}$ whenever the true (but unknown) ratio of the largest to the second largest of the population probabilities is equal to or greater than $\theta^{\ast}$." A single-sample procedure which meets the requirement is proposed. The heart of the procedure is the proper choice of $N$, the number of trials. Two methods of determining $N$ are described: the first is exact and is to be used when $N$ is small; the second is approximate and is to be used when $N$ is large. Tables and sample calculations are provided.

Certain $\psi^2$ statistics were defined by Good in [8] and he stated that there is a "strong analogy" between certain functions of these statistics and some given likelihood ratio (LR) statistics appropriate for testing hypotheses concerning the order of a Markov chain. He also indicated that the analogy held when the hypothesis of "perfect randomness" is true. The present author has indicated in [12] that, for the $\psi^2$ statistics in [8], this analogy (i.e., the asymptotic equivalence of the corresponding statistics) does not hold under some more general conditions when the "perfect randomness" hypothesis is not true. It will be seen herein that certain functions of a modified form of the $\psi^2$ statistics are asymptotically equivalent to certain LR statistics in the more general case when the hypothesis $H(P_m)$ that the positively regular Markov chain (see [2]) is governed by a completely specified system $P_m$ of $m$th order transition probabilities is true. Also, certain functions of a different modified form of the $\psi^2$ statistics will be seen to be asymptotically equivalent to certain LR statistics in the case when the hypothesis $H_m$ that the positively regular Markov chain is of order $m$ is true. These results are helpful in determining the asymptotic distributions of various statistics and the null hypotheses that can be tested with a given statistic. For example, if a given statistic $G$ is asymptotically equivalent, under $H(P_1),$ to the LR statistic $L$ for testing the null hypothesis $H(P_1)$ within the alternate hypothesis $H_2$, then the asymptotic distribution, under $H(P_1),$ of $G$ will be $\chi^2$ with a known number of degrees of freedom (i.e., with a known expectation); $G$ can be used directly to test $H(P_1)$ within $H_2$ (if $G$ is sensitive to these hypotheses), although the asymptotic distribution, under $H_2,$ of $G$ may differ, in a certain sense, from that of $L$ (see Section 6 in [1]). However, if a given statistic $\Delta G$ is asymptotically equivalent, under $H(P_1),$ to the LR statistic $\Delta L$ for testing the null hypothesis $H_1$ within the alternate hypothesis $H_2$ (i.e., if there is an "ostensible analogy" between $\Delta G$ and $\Delta L$), but this asymptotic equivalence does not hold under some more general conditions (e.g., under $H_1$), then $\Delta G$ can not be used to test $H_1$ within $H_2$; the asymptotic distribution, under $H(P_1),$ of $\Delta G$ will be $\chi^2,$ but the asymptotic distribution, under the null hypothesis $H_1,$ will not be $\chi^2,$ and furthermore the expectation, under $H_1,$ of $G$ can approach infinity (see [15]). The present author has indicated in [15] that certain functions of a modified form of the $\psi^2$ statistics, which were investigated by Stepanow in [18] and which are computed for a specified $P_1$, are asymptotically equivalent, under $H(P_1),$ to certain LR statistics, but that they will not be equivalent under $H_1.$ Although it is stated in [18] that the results presented there can be applied to the solution of the problem of testing the null hypothesis $H_1,$ it is shown in [15] that none of the statistics in [18] can be used directly to test this composite hypothesis. In the present paper, it will be seen that a statistic based on a different modified form of $\psi^2,$ as well as certain other statistics described herein, will be asymptotically equivalent, under $H_1,$ to a certain LR statistic and can be used to test the null hypothesis $H_1.$ The $\psi^2$ statistics defined by Good in [10] are more general than those given in [3], [8], [18]. Besides studying the relation between these statistics and the LR statistics, we shall also discuss certain conjectures proposed in [10] concerning the asymptotic distributions of these statistics, which were investigated by Billingsley [4] for the cases $H_0$ and $H_1$ (the author mentions that a more general result for $H_m(m \geqq 0)$ can be obtained using similar methods) and independently, using different methods, by the present author [14] for the case $H_m(m \geqq 0)$ when the transition probabilities are all positive (this author also mentions that a more general result can be obtained by similar methods). The $\psi^2$ and LR statistics defined in [10] and [8], as well as some related statistics developed in the present paper, will be generalized further herein, and the asymptotic distributions of these generalized statistics will be investigated. This investigation leads to generalizations of the asymptotic distributions obtained by Good [8], Billingsley [3] [4], and the present author [13] [14], and it helps to clarify the relation between the various statistics. The different asymptotically equivalent forms of various statistics presented here make it possible for the statistician to choose whichever form he finds preferable both from the computational point of view and also from some other viewpoints (see [1], [5], [11]).

Gnedenko and Korolyuk [1] have pointed out that the exact distribution of the Kolmogorov-Smirnov $D$-statistic can be obtained explicitly by solving a certain double-boundary random walk problem, which, in turn, is solved by the principle of reflection. This principle is employed here in what is believed to be a new way to derive Gnedenko's and Korolyuk's result.

In this paper, using the heuristic union-intersection principle [4], two tests are proposed, and the associated simultaneous confidence bounds on parametric functions which are measures of a certain type of departure from the respective null hypotheses are obtained. The first test is for the equality of $(k + 1)$ variances $(k \geqq 2)$ of $(k + 1)$ univariate normal populations, wherein we choose one of the variances as a standard (of course, unknown), and compare the other $k$ variances with it. The alternative to the hypothesis is that not all the $k$ variances are equal to the standard one. The proposed test may be called the simultaneous variance ratios test. The well-known Hartley's $F_{\max}$ test [2] for the case of equal sample sizes is not equivalent to the present test even when all samples are of the same size since the alternatives in the two cases are different. In the alternative in Hartley's test, aside from the inequality of the $k$ variances to the standard one, the mutual inequality of the $k$ variances also plays an important role. The second test proposed in this paper, is a multivariate extension of the first. This paper also considers the distribution problems that arise in connection with both the tests. The nonavailability of tables at the moment makes the immediate practical application of the tests and the associated confidence bounds not possible. Sections 1, 2, and 3 deal with the univariate problem and Sections 4 and 5 deal with the multivariate extensions.

The notion of best asymptotically normal estimates--BAN estimates for short--was introduced by Neyman [8] in the multinomial case. Applications have been made in biological problems, notably in bio-assay [2], [4], [5]. Generalizations of Neyman's work have been made by Barankin and Gurland [1], Chiang [3] and Ferguson [5]. The usual theory of BAN estimates requires differentiability of the estimates, and imposes rather strong conditions on certain functions given in advance (the functions $\zeta$ and $\Sigma$ of Section 3). In this note a different definition of BAN estimates is made which does not require differentiability, at the same time relaxing the conditions on $\zeta$ and $\Sigma,$ whereas in essence all important theorems in the theory of BAN estimates are retained.

Joint intervals of bounded confidence are suggested for the medians of a bivariate population with continuous marginal distributions. The two intervals are of the classic type based on sample order statistics.

Suppose the observed random variable $X$ is the sum of two independent random variables $Z$ and $Y,$ where $Z$ has a normal distribution with zero expectation and a known variance, and $Y$ has a distribution function, say $G(y)$, which is completely unknown. Then the distribution function of $X$ may be written as \begin{equation*}\tag{1.1} F(x) = \frac{1}{\sqrt{2\pi\sigma}}\int^\infty_{-\infty}G(y)\exp\big\lbrack - \frac{(x - y)^2}{2\sigma^2}\big\rbrack dy,\end{equation*} where $F(x)$ and $G(y)$ are unknown. We consider here the problem of estimating $G(y)$ from a sample $x_1, x_2, \cdots x_n.$ Such a problem may arise if, for example, each $x_i$ represents a serum cholesterol determination on one human being randomly selected from some population. Then each $x_i$ may be thought of as the true cholesterol value for that person, plus an "instrumental error" introduced by the complex chemical analysis. We may wish to "correct" for the instrumental error, so to speak, by estimating the distribution of true cholesterol levels in the population. The maximum likelihood and minimum distance principles do not seem to yield estimators which may be expressed as explicit, more or less easily computable functions of the sample values. We present such an estimator, which is consistent at every continuity point of $G(y).$ (We consider only continuity points throughout the paper.) The estimator is constructed by first exhibiting an inversion formula for $G(y)$ in terms of the derivatives of $F(x),$ and then replacing the derivatives by the difference quotients of the empiric distribution function $F_n(x).$ The asymptotic mean square error of the estimator is derived, and a rough rule is suggested for deciding when it is worthwhile to compute an estimate. The fact that the estimator is still consistent under certain kinds of dependence between $Z$ and $Y$ is indicated. Finally, some comments are made on the relationship between the present estimator and one derived by Eddington.

Some of the sampling methods and the methods of estimation usually employed in sample surveys are considered in terms of loss and risk functions. The loss function is taken as the sum of two components, one proportional to the square of the error of the estimate and the other proportional to the cost of obtaining the sample. Consideration is given to the problem of the allocation of the total sample size and only non-sequential estimates are discussed. As the loss function is convex and of finite expectation in each case, only non-randomized estimates are considered, since Hodges and Lehmann [5] have shown that under these conditions the class of non-randomized estimates is essentially complete. Only simple random sampling and stratified sampling methods are discussed in this part, the ratio, regression and sub-sampling methods will be discussed in subsequent parts.

This paper examines the loss of power when using tests based on the assumption that the variable being sampled has a "complete" normal distribution when in fact the distribution is a "truncated" one. The cases considered here are for small sample sizes and "symmetric" truncation, while the hypothesis considered is the one-sided testing for the mean of a normal distribution. Some tables are computed and it appears that an appreciable loss occurs only in the size of the test. The loss in power is found to decrease very rapidly with the distance of the alternative value of the mean from the one tested and also with the distance of the truncation from the mean.

B. McMillan and J. Riordan in [1] derived the generating function for the probability distribution of the number of items completed before absorption in a moving single server problem in two special cases. Through an analogy to the work of L. Takacs [2] on busy period problems for a simple queue, McMillan and Riordan postulated a nonlinear integral equation relation for the generating function. In this note the validity of this relation is proved in general by exploiting the analogy more fully, and the generating function in the two special cases is obtained directly from the integral equation. A similar functional relation is established for the Laplace-Stieltjes transform of the distribution of time until absorption, and the transform is obtained for the two special cases.