In this paper recent results on the lower bound to the variance of unbiased estimates have been brought together. Some of them have been extended to sequential estimates and the others have been improved to some extent. In the last section a general method for generating a system of orthogonal polynomials with respect to a certain class of weight functions is obtained together with a result on the conditions under which the class of unbiased estimates formed by all functions of an unbiased estimate consists of just one element.

For two types of non-parametric hypotheses optimum tests are derived against certain classes of alternatives. The two kinds of hypotheses are related and may be illustrated by the following example: (1) The joint distribution of the variables $X_1, \cdots, X_m, Y_1, \cdots, Y_n$ is invariant under all permutations of the variables; (2) the variables are independently and identically distributed. It is shown that the theory of optimum tests for hypotheses of the first kind is the same as that of optimum similar tests for hypotheses of the second kind. Most powerful tests are obtained against arbitrary simple alternatives, and in a number of important cases most stringent tests are derived against certain composite alternatives. For the example (1), if the distributions are restricted to probability densities, Pitman's test based on $\bar y - \bar x$ is most powerful against the alternatives that the $X$'s and $Y$'s are independently normally distributed with common variance, and that $E(X_i) = \xi, E(Y_i) = \eta$ where $\eta > \xi$. If $\eta - \xi$ may be positive or negative the test based on $|\bar y - \bar x|$ is most stringent. The definitions are sufficiently general that the theory applies to both continuous and discrete problems, and that tied observations present no difficulties. It is shown that continuous and discrete problems may be combined. Pitman's test for example, when applied to certain discrete problems, coincides with Fisher's exact test, and when $m = n$ the test based on $|\bar y - \bar x|$ is most stringent for hypothesis (1) against a broad class of alternatives which includes both discrete and absolutely continuous distributions.

A method is given for estimating the coefficients of a single equation in a complete system of linear stochastic equations (see expression (2.1)), provided that a number of the coefficients of the selected equation are known to be zero. Under the assumption of the knowledge of all variables in the system and the assumption that the disturbances in the equations of the system are normally distributed, point estimates are derived from the regressions of the jointly dependent variables on the predetermined variables (Theorem 1). The vector of the estimates of the coefficients of the jointly dependent variables is the characteristic vector of a matrix involving the regression coefficients and the estimate of the covariance matrix of the residuals from the regression functions. The vector corresponding to the smallest characteristic root is taken. An efficient method of computing these estimates is given in section 7. The asymptotic theory of these estimates is given in a following paper [2]. When the predetermined variables can be considered as fixed, confidence regions for the coefficients can be obtained on the basis of small sample theory (Theorem 3). A statistical test for the hypothesis of over-identification of the single equation can be based on the characteristic root associated with the vector of point estimates (Theorem 2) or on the expression for the small sample confidence region (Theorem 4). This hypothesis is equivalent to the hypothesis that the coefficients assumed to be zero actually are zero. The asymptotic distribution of the criterion is shown in a following paper [2] to be that of $\chi^2$.

Order statistics are used to derive significance tests for the population median which are valid under very general conditions. These tests are approximately as powerful as the Student $t$-test for small samples from a normal population. Also the application of a test requires very little computation. Thus the tests derived compare very favorably with the $t$-test for small sets of observations. Applications of these order statistic tests to certain well known statistical problems are given in another paper [1].

A compounding technique first used to produce random binary digits is generalized and extended to other number systems. Formulae for the rate of convergence of probabilities to the desired values are derived. The method is extended to the production of random digits with fixed but unequal probabilities. Numerical results are presented in summary form together with results of tests applied to a set of random digits produced by the method.

A statistic useful for detecting deviations from the Hardy-Weinberg equilibrium in population genetics is discussed. Both exact and asymptotic distributions are given and a special case where there is misclassification is discussed. The distribution obtained also arises from a certain card matching problem.

Consider the following problem. Particles are distributed over unit areas in such a way that the number of particles to be found in such areas is a random variable following the law of Poisson, with $\nu$ equal to the expected number of particles per unit area. Furthermore, the particles themselves are assumed to vary in magnitude according to a size distribution specified (independently of the particular unit area chosen) by a d.f. $F(x)$ defined over some interval $a \leq x \leq b$, with $F(a) = 0$ and $F(b) = 1$. The problem is to find the distribution of the smallest, largest, or more generally the $n$th smallest or $n$th largest particle in randomly chosen unit areas. The problem as stated is not completely specified. To specify the distribution of smallest or largest particles in a unit area one must give a rule for dealing with those areas which contain no particles at all. More generally, in the case of the distribution of the $n$th smallest or $n$th largest particle, one must give a rule for dealing with those areas which contain $(n - 1)$ or fewer particles. There are at least two possible alternatives. One alternative is to omit none of the areas from consideration by setting up the following rule: if no particles are found in a given unit area then this area will be considered as one for which the smallest size particle is $x = b$ and for which the largest size particle is $x = a$. More generally, if $(n - 1)$ or fewer particles are found in a given unit area then this area will be considered as one for which the $n$th smallest size particle is $x = b$ and for which the $n$th largest size particle is $x = a$. A second alternative is to restrict attention to those areas which contain at least one particle (in the case of the distribution of smallest or largest values) or at least $n$ particles (in the case of the distribution of the $n$th smallest or $n$th largest particle). In other words, this means finding the relevant conditional distribution. From the point of view of the application of the theory of extreme values to fracture problems, there are some situations where the first model and other situations where the second model is the more appropriate in describing the phenomenon under investigation. In this paper section 2 will be devoted to a derivation of the distributions associated with the first alternative; in section 3 the conditional distributions will be described briefly.

Since the ordinary tests for independence in contingency tables use test criteria whose distributions depend on unknown parameters, the justification for the tests is usually made either by an appeal to asymptotic theory or by interpreting the tests as conditional tests. The latter approach employs the conditional distribution of the cell frequencies given the marginal totals, and was first described by Fisher [1]. The purpose of the present note is to show how these tests may be regarded as unconditional tests even though the parameters are unknown by augmenting the test criterion to include estimates of the unknown parameters. We present no new tests, merely a new setting for the old tests which seems to put them in a little better light.