In two recent papers, Paulson [1] and Mosteller [2] have called attention to several unsolved problems in $k$-sample theory. A problem which is typical of the ones considered in this paper is as follows. Let $\pi_1, \pi_2, \cdots, \pi_k$ be a set of normal populations, $\pi_i$ having an unknown mean $m_i$ and variance $\sigma^2, G(x, \theta_i)$ being the distribution function which characterizes $\pi_i$. Samples of equal size are drawn from each population, $\bar X_i$ being the sample means, and $S^2$ the estimate of $\sigma^2$ obtained. The problem is to construct a suitable decision rule $d = d(\{\bar X_i\}; S^2)$ to select one or more populations, the object being to minimize the expected value of the random distribution function $G(x \mid s(d)) = \sum^k_{i=1} Z_i(d) \cdot G(x, \theta_i) / \sum^k_{i=1} Z_i(d),$ where $Z_i(d) = 1$ if $\pi_i$ is selected by $d$, and $= 0$ otherwise. It is shown that under the restriction of impartial decision, the rule $d_k =$ "Always select only the population corresponding to the greatest $\bar X_i$" cannot be improved, no matter what $x$ or the true parameter values may be. It follows (i) that $d_k$ is the uniformly best decision rule in the class of impartial decision rules for all weight functions of type $W = \max_i \{m_i\} - \big(\sum^k_{i=1} z_im_i / \sum^k_{i=1} z_i\big)$, and (ii) that the customary $F$ and $t$ tests of analysis of variance are not relevant to the problem. This result is an application of Theorem 1 which applies to a number of similar problems concerning $k$ populations, especially when the populations admit sufficient statistics for their parameters. Two examples of statistical applications are given in Section 6.

Recently, in a series of papers, Girshick, Mosteller, Savage and Wolfowitz have considered the uniqueness of unbiased estimates depending only on an appropriate sufficient statistic for sequential sampling schemes of binomial variables. A complete solution was obtained under the restriction to bounded estimates. This work, which has immediate consequences with respect to the existence of unbiased estimates with uniformly minimum variance, is extended here in two directions. A general necessary condition for uniqueness is found, and this is applied to obtain a complete solution of the uniqueness problem when the random variables have a Poisson or rectangular distribution. Necessary and sufficient conditions are also found in the binomial case without the restriction to bounded estimates. This permits the statement of a somewhat stronger optimum property for the estimates, and is applicable to the estimation of unbounded functions of the unknown probability.

Let us consider a situation where only the $r$ smallest values of a sample of size $n$ are available. This paper investigates the case where $n$ is large and $r$ is of the form $pn + O(\sqrt {n})$. Properties of some well known non-parametric point estimates, confidence intervals and significance tests for the 100p% point of the population are investigated. If the sample is from a normal population, these non-parametric estimates and tests have high efficiencies for small values of $p$ (at least 95% if $p \leqq 1/10)$. The other results of the paper are restricted to the special case of a normal population. Asymptotically "best" estimates and tests for the population percentage points are derived for the case in which the population standard deviation is known. For the case in which the population standard deviation is unknown, asymptotically most efficient estimates and tests can be obtained for the smaller population percentage points by suitable choice of $p$ and $O(\sqrt {n})$. The results derived have application in the field of life testing. There the variable associated with an item is the time to failure and the $r$ smallest sample values can be obtained without the necessity of obtaining the remaining values of the sample. By starting with a larger number of units but stopping the experiment when only a small percentage of the units have "died", it is often possible (using the results of this paper) to obtain the same amount of "information" with a substantial saving in cost and time over that which would be required if a smaller number of units were used and the experiment conducted until all the units have "died". Jacobson called attention to applications of this type in [1].

Subject to certain restrictions, a characterization of unbiased estimates with minimum variance is obtained. For two fairly broad classes of problems, solutions are given which are more readily applicable. These are used to obtain such estimates in some particular cases. The applicability of the results to problems of sequential estimation is pointed out. The problem of unbiased estimation is not at present of much practical importance, but is of some theoretical interest and has been treated by many statisticians. Also, the method used in this paper may be applicable to other problems in statistics.

Given a chance vector $\mathbf{X}$ with distribution function $F(\mathbf{X}, \theta_T)$, where $\mathbf{theta}_T$ denotes the true unknown parameter vector, a broad class of estimates of $\mathbf{theta}_T$ is derived which is shown to be identical with the class of all consistent estimates of $\mathbf{\theta}_T$. A sub-class is obtained each member of which has the following properties: a.) Its construction depends upon the solution of an equation involving a single vector function of the parameter vector $\theta$ and the members of a sequence $\{\mathbf{X}_n\}$ of independent and identically distributed chance vectors; b.) the estimate so obtained converges almost certainly to $\mathbf{\theta}_T$; c.) it is a symmetric function of the members of the sequence $\{\mathbf{X}_n\}$. In order to obtain this subclass it is postulated that a function of $\mathbf{X}$ and $\mathbf{\theta}$ exists (continuous in $\mathbf{\theta}$ for a certain neighborhood of the true parameter $\mathbf{\theta}_T$ and existing for each $\mathbf{X}$ in a subset of the sample space) which satisfies a Lipschitz condition in $\mathbf{\theta}$. In particular if a density function $f(\mathbf{X}, \theta_T)$ exists satisfying certain conditions, the consistency of the maximum likelihood estimate can be established under regularity conditions quite different from those usually assumed [1]. This is not to be interpreted as a weakening of the usual regularity conditions but rather as an extension of the class of consistent likelihood estimates obtained under the usual regularity conditions.

This paper is in continuation of the author's first two papers [1] and [2]. In this paper a method is described by which it is possible to derive the distribution of the sum of roots of a certain determinantal equation under the condition that $m = 0$. This condition implies, when the results are applied to canonical correlations, that the numbers of variates in the two sets differ by unity. The distributions for the sum of roots under this condition have been obtained for $l$ = 2, 3 and 4 and are given in this paper. This paper also derives the moments of these distributions.

Deltheil ([1], pp. 114-120) has considered the distribution of distance in an $n$-dimensional hypersphere. In this paper I put his results (17) in a more compact form (16); and I investigate in greater detail the asymptotic form of the distribution for large $n$, for which the rather surprising result emerges that this distance is almost always nearly equal to the distance between the extremities of two orthogonal radii. I came to study this distribution by the need to compute a doubly-threefold integral, which measures the damage caused to plants by the presence of radioactive tracers in their fertilizers; for the distribution affords a method of evaluating numerically certain multiple integrals. I hope to describe elsewhere this application of the theory.