The general existence of minimax strategies and other important properties proved in the theory of statistical decision functions (e.g., [3]) and the theory of games (e.g., [5]) depends upon the convexity of the space of decision functions and the convexity of the space of strategies. This convexity can be obtained by the use of randomized decision functions and mixed (randomized) strategies. In Section 2 of the present paper the authors state the extension (first announced in [1]) of a measure theoretical result known as Lyapunov's theorem [2]. This result is applied in Section 3 to the statistical decision problem where the number of distributions and decisions is finite. It is proved that when the distributions are continuous (more generally, "atomless," see footnote 7 below) randomization is unnecessary in the sense that every randomized decision function can be replaced by an equivalent nonrandomized decision function. Section 4 extends this result to the case when the decision space is compact. Section 5 extends the results of Section 3 to the sequential case. Sections 6 and 7 show, by counterexamples, that the results of Section 3 cannot be extended to the case of infinitely many distributions without new restrictions. Section 8 gives sufficient conditions for the elimination of randomization under maintenance of $\epsilon$-equivalence. Section 9 concludes with a restatement of the results in the language of the theory of games.

This paper is concerned with the problem of making a decision on the basis of a sequence of observations on a random variable. Two loss functions, each depending on the distribution of the random variable, the number of observations taken, and the decision made, are assumed given. Minimax problems can be stated for weighted sums of the two loss functions, or for either one subject to an upper bound on the expectation of the other. Under suitable conditions it is shown that solutions of the first type of problem provide solutions for all problems of the latter types, and that admissibility for a problem of the first type implies admissibility for problems of the latter types. Two examples are given: Estimation of the mean of a random variable which is (1) normal with known variance, (2) rectangular with known range. The resulting minimax estimates are, with a small class of exceptions, proved admissible among the class of all procedures with continuous risk functions. The two loss functions are in each case the number of observations, and an arbitrary nondecreasing function of the absolute error of estimate. Extensions to a function of the number of observations for the first loss function are indicated, and two examples are given for the normal case where the sample size can or must be randomised among more than a consecutive pair of integers.

A case of nonregular estimation arises in attempting to estimate a single unknown parameter, $\theta$, in the probability distribution of a single chance variable in which one or both of the extremities of the range of the distribution are functions of the unknown parameter. The case treated in this paper is the one in which a probability density of exponential type exists. When one extremity alone of the range depends non-trivially upon $\theta$, a necessary and sufficient condition is given in order that a single order statistic be a sufficient statistic for $\theta$. In this case conditions are given for the existence of a unique unbiased estimate of $\theta$ possessing minimum variance uniformly in $\theta$. In the case in which both extremities of the range depend upon $\theta$, a necessary and sufficient condition is given that the smallest and largest order statistics constitute a set of sufficient statistics for $\theta$. In this case Pitman [1] has shown that a single sufficient statistic exists if one extremity of the range is a monotone decreasing function of the other extremity. It is shown that under the above condition a unique unbiased estimate exists possessing minimum variance. Moreover a surmise of Pitman is proved that only under this condition does a single sufficient statistic exist. When a single sufficient statistic does not exist, an unbiased estimate of a known function of $\theta$ is obtained which has less variance than any analytic function of the set of sufficient statistics for $\theta$.

In this paper we shall consider the exact distribution of Wald's classification statistic $V$ in the univariate case, some theoretical approximations in various multivariate cases, and an empirical distribution in a particular multivariate case. We shall also draw some conclusions as to the potential usefulness of the statistic $V$ and the work which remains to be done.

Ratios of the form $(x_n - x_{n-j})/(x_n - x_i)$ for small values of $i$ and $j$ and $n = 3, \cdots, 30$ are discussed. The variables concerned are order statistics, i.e., sample values such that $x_1 < x_2 < \cdots < x_n$. Analytic results are obtained for the distributions of these ratios for several small values of $n$ and percentage values are tabled for these distributions for samples of size $n \leqq 30$.

The following lemma proved by Neyman and Pearson [1] is basic in the theory of testing statistical hypotheses: LEMMA. Let $f_1(x), \cdots, f_{m+1}(x)$ be $m + 1$ Borel measurable functions defined over a finite dimensional Euclidean space $R$ such that $\int_R |f_i(x)|dx < \infty (i = 1, \cdots, m + 1)$. Let, furthermore, $c_1, \cdots, c_m$ be $m$ given constants and $\mathcal{S}$ the class of all Borel measurable subsets $S$ of $R$ for which (1.1) $\int_S f_i(x) dx = c_i \\ (i = 1, \cdots, m)$. Let, finally, $\mathcal{S}_0$ be the subclass of $\mathcal{S}$ consisting of all members $\mathcal{S}_0$ of $\mathcal{S}$ for which (1.2) $\int_{S_0} f_{m + 1}(x) dx \geqq \int_S f_{m+1}(x) dx \text{for all S in} \mathcal{S}$. If $S$ is a member of $\mathcal{S}$ and if there exist $m$ constants $k_1, \cdots, k_m$ such that (1.3) $f_{m + 1}(x) \geqq k_1f_1(x) + \cdots + k_mf_m(x) \text{when} x \epsilon S$, (1.4) $f_{m + 1}(x) \leqq k_1f_1(x) + \cdots + k_mf_m(x) \text{when} x \not\epsilon S$, then $S$ is a member of $\mathcal{S}_0$. The above lemma gives merely a sufficient condition for a member $S$ of $\mathcal{S}$ to be also a member of $\mathcal{S}_0$. Two important questions were left open by Neyman and Pearson: (1) the question of existence, that is, the question whether $\mathcal{S}_0$ is non-empty whenever $\mathcal{S}$ is non-empty; (2) the question of necessity of their sufficient condition (apart from the obvious weakening that (1.3) and (1.4) may be violated on a set of measure zero). The purpose of the present note is to answer the above two questions. It will be shown in Section 2 that $\mathcal{S}_0$ is not empty whenever $\mathcal{S}$ is not empty. In Section 3, a necessary and sufficient condition is given for a member of $\mathcal{S}$ to be also a member of $\mathcal{S}_0$. This necessary and sufficient condition coincides with the Neyman-Pearson sufficient condition under a mild restriction.

Two estimators of the probability of falling into the zero class are compared, for a family of populations related to Poisson populations. The first estimator, $\epsilon_1$, is based on the observed proportion in the zero class; the second, $\epsilon_2$, would be the maximum likelihood estimator if the underlying distribution were Poisson. From a practical point of view each estimator possesses its own peculiar advantages. $\epsilon_1$ has the advantage that the detailed distribution among the non-zero classes need not be examined. $\epsilon_2$ has the advantage that only the mean of the observations is needed, the distribution among the various classes not being required. The relative importance of these advantages will naturally vary according to the situations in which the estimators are to be used. An arbitrary measure of relative accuracy, the mean square error ratio, is used. On this basis $\epsilon_2$ is superior to $\epsilon_1$ for all sample sizes (greater than one) if the population distribution is Poisson. Provided the sample size is not too large $\epsilon_2$ may still be superior to $\epsilon_1$ when the population distribution deviates to a moderate extent from Poisson form. A third estimator $\epsilon_3$, which is a modification of $\epsilon_2$ and is unbiased, provided the population is Poisson, may be preferred to $\epsilon_2$ unless $p$ exceeds about 0.45. Its properties vis-a-vis $\epsilon_1$ probably differ little from those of $\epsilon_2$.

The problem of comparing the proportionality of covariance matrices often arises in genetic experiments. Knowledge of nonproportionality of covariance matrices is useful in selection work and in genetic interpretations. In developing a test of significance for this contrast, the likelihood ratio criterion was used. Likelihood ratio tests were obtained for two sets and for three sets of independent variance-covariance matrices. The test for $r$ independent covariances was indicated and some unsolved problems were cited.

In this note the author points out that in the case of stationary Guassian Markov process, i.e., autoregressive stochastic process, we can test the serial correlation coefficients by a method based on normal regression theory. Particularly, in the case of simple Markov process, we can find the confidence limits for its autocorrelation coefficient. In this method, so far as random variables are concerned, not all the information in the original data is used, with a consequence reduction of degrees of freedom. However, the other part of information is introduced as parameters in the distribution functions of random variables and in the statistic useful for tests.

Let $x_1 < x_2 M \cdots < x_n$ and $y_1 < y_2 < \cdot < y_m$ be the ordered results of two random samples from populations having continuous cumulative distribution functions $F(x)$ and $G(x)$ respectively. Let $S_n(x) = K/n$ when $k$ is the number of observed values of $X$ which are less than or equal to $x$, and similarly let $S'_m(y) = j/m$ where $j$ is the number of observed values of $Y$ which are less than or equal to $y$. The statistic $d = \max | S_n(x) - S'_m(x) |$ can be used to test the hypothesis $F(x) \equiv G(x)$, where the hypothesis would be rejected if the observed $d$ is significantly large. The limiting distribution of $d \sqrt{mn}{m + n}$ has been derived [1] and [4], and tabled [5]. In this paper a method of obtaining the exact distribution of $d$ for small samples is described, and a short table for equal size samples is included. The general technique is that used by the author for the single sample case [2]. There is a lower bound to the power of the test against any specified alternative, [3]. This lower bound approaches one as $n$ and $m$ approach infinity proving that the test is consistent.