Many problems in the theory of probability and statistics can be solved by evaluating coefficients in generating function, or, for continuous differentiable distributions, by an analogous process with Laplace or Fourier transforms. As pointed out for example by H. E. Daniels [2], these problems can often be solved by asymptotic series derived by the saddle-point method from integrals containing a large parameter. Daniels gave a form of saddle-point theorem that is convenient for applications to probability and statistics. In the present paper we extend the theorem in various directions and give some applications to distributions connected with the multinomial distribution, especially to the distribution of $\chi^2$ and to the distribution of the maximum entry in a multinomial distribution.

In genetics, stochastic processes arise at all levels of organization ranging from subunits of the gene to natural populations. Types of stochastic processes involved are also diverse. In the present paper, the following five topics have been selected for mathematical discussion and new results are presented: (1) Random assortment of subunits of a gene. (2) Senescence in paramecium due to random assortment of chromosomes. (3) Process of natural selection in a finite population (interaction between selection and random genetic drift). (4) Chance of fixation of mutant genes. (5) Population structure and evolution. Finally it is pointed out that new mathematical techniques will be needed for a satisfactory treatment of Wright's theory of evolution.

The comparison of the sensitivities of experiments using different scales of measurement or different experimental techniques can be effected through a comparison of noncentral variance ratios. The distribution of the ratio of two noncentral variance ratios is obtained and its properties are discussed. Based on this distribution, tests of hypotheses on the parameters on noncentrality of two noncentral variance-ratio distributions are developed. It is shown that the distribution of the ratio of two noncentral variance ratios may be approximated adequately by the distribution of the ratio of two central variance ratios with appropriately adjusted degrees of freedom. A table for use in applications of the latter distribution is given for one-sided tests at the 5% level of significance. Through the association of the distribution of the multiple correlation coefficient in regression models with that of the noncentral variance ratio, it was also possible to develop test procedures on multiple correlation coefficients. Much of the discussion in this paper is on comparisons of similar experiments in the sense that variance ratios with the same degrees of freedom are compared. However, it is shown how these results may be generalized for comparisons of dissimilar experiments.

Let $X, Y$ be independent random variables with continuous cumulative probability functions and let $$p = \mathrm{Pr}\{Y < X\}.$$ For the variance of the Mann-Whitney statistic $U,$ upper and lower bounds are obtained in terms of $p$, for the case of any $X$ and $Y$ as well as for the case of stochastically comparable $X, Y$. The results for the case of stochastic comparability are new, while the inequalities in the case of arbitrary $X, Y$ have either been obtained by van Dantzig or are a consequence of other inequalities due to van Dantzig.

Using a theorem due to de Bruijn, van Aardenne-Ehrenfest, C. A. B. Smith and Tutte concerning the number of circuits in oriented linear graphs, an expression is found for the probability of a specified frequency count of $m$-tuples in a circular sequence where the $n$-tuple $(n < m)$ count is given. The corresponding result for linear sequences can be deduced--see [14]. The result is valid for stationary Markovity of any order up to and including the $(n - 1)$-st. A method of deriving asymptotic distributions is indicated, and a few additional observations made concerning the distribution of pairs in a circular array.

The statistic $y = (x_{\lbrack p\rbrack} - x)/s_\nu$ is studied where $x_{\lbrack p\rbrack}$ is the maximum of $p$ normal independent chance variables with common mean and common unknown variance $\sigma^2, x$ is another independent normal chance variable with the same mean and the same variance $\sigma^2,$ and $s^2_\nu$ (distributed as $\sigma^2\chi^2_\nu/\nu$ with $\nu$ degrees of freedom) is an estimate of the common variance which is independent of each one of the above $p + 1$ chance variables. Several different methods are proposed and studied for computing the probability integral of $y$ and percentage points of $y$; in addition, a method for computing percentage points without first computing the probability integral of $y$ is considered. A table of (upper) percentage points of $y$ is given as Table I at the end of the paper. Applications of the statistic $y$ to several ranking and selection problems are mentioned in Section 2. Moments of $y$ are given in Section 3. In Section 7 it is shown that Table I can be used to obtain an approximation and bounds to the percentage points of a related statistic.

The distribution of the sum of $n$ random coplanar unit vectors and of a given component of the sum has been discussed by many authors, who have shown that each distribution can be approximated in series that are asymptotically normal. But the difficult question of the usefulness of these approximations for finite $n$--in particular for small $n$--has not been exhaustively treated. Accordingly, this paper reexamines some analyses of Pearson's series for the vector sum, presents corresponding series for a component, and examines the accuracy of the latter series.

Suppose that the integers $1, 2, \cdots, N$ are randomly distributed among $k$ distinguishable classes with equal probability and without restrictions. It is natural to denote the class sums by $s_1, s_2, \cdots, s_k$ and largest of these by $S$. A generating function is obtained for the upper half of the range of $S$, namely $\frac{1}{4}N(N - 1) \leqq S \leqq \frac{1}{2} N(N - 1)$. For $k \leqq 6$, this is shown to provide the usual percentage points for $N$ up to and beyond 10. Tables of 5% and 1% points are provided for $k = 2, 3, 6$ and $N = 1(1)10$. For $k = 2$, the distribution is that of Wilcoxon's paired sample test [3]. This suggests the application of $k = 6$ to the six possible orders of three responses. This is a possible procedure but the peculiarities of its power are such that its use is not recommended. However, when three treatments with a natural order are examined in randomized blocks, a significance procedure can be based on the same distribution which is specifically sensitive to average responses in either exactly the same or exactly the opposite order as the treatments. 5% and 1% levels are given for $N = 1(1)10$. The procedure may be promising. The basic distribution used here is inappropriate for situations, such as analysis of variance of ranks, where the number of ranks in each class is restricted, as by being the same in all classes.

Important properties of an experimental design, including the analysis of variance appropriate to it, are revealed by analysing the structure of an algebra generated by the relationships between the experimental units of the design. As an illustration, the relationship algebra of balanced incomplete blocks is analysed in detail.

Let $M = \| m_{ij} \|$ be a $4 \times 4$ irreducible aperiodic Markov matrix such that $h_1 \neq h_2, h_3 \neq h_4$, where $h_i = m_{i1} + m_{i2}$. Let $x_1, x_2, \cdots$ be a stationary Markov process with transition matrix $M$, and let $y_n = 0$ when $x_n = 1$ or 2, $y_n = 1$ when $x_n = 3 \text{or} 4$. For any finite sequence $s = (\epsilon_1, \epsilon_2, \cdots, \epsilon_n)$ of 0's and 1's, let $p(s) = \mathrm{Pr}\{y_1 = \epsilon_1, \cdots, y_n = \epsilon_n\}$. If \begin{equation*}\tag{1}p^2(00) \neq p(0)p(000) \text{and} p^2(01) \neq p(1)p(010),\end{equation*} the joint distribution of $y_1, y_2, \cdots$ is uniquely determined by the eight probabilities $p(0), p(00), p(000), p(010), p(0000), p(0010), p(0100), p(0110)$, so that two matrices $M$ determine the same joint distribution of $y_1, y_2, \cdots$ whenever the eight probabilities listed agree, provided (1) is satisfied. The method consists in showing that the function $p$ satisfies the recurrence relation \begin{equation*}\tag{2}p(s, \epsilon, \delta, 0) = p(s, \epsilon, 0)a(\epsilon, \delta) + p(s, \epsilon)b(\epsilon, \delta)\end{equation*} for all $s$ and $\epsilon = 0$ or 1, $\delta = 0$ or 1, where $a(\epsilon, \delta), b(\epsilon, \delta)$ are (easily computed) functions of $M$, and noting that, if (1) is satisfied, $a(\epsilon, \delta)$ and $b(\epsilon, \delta)$ are determined by the eight probabilities listed. The class of doubly stochastic matrices yielding the same joint distribution for $y_1, y_2, \cdots$ is described somewhat more explicitly, and the case of a larger number of states is considered briefly.

Certain functions of the sample spacings are shown to converge stochastically as the sample size increases. This leads to certain convenient tests of fit which are consistent against wide classes of alternatives.

For the class of distribution functions given by $$dP(x, \theta) = \exp \lbrack r(\theta)A(x) + s(\theta)B(x)\rbrack dw(x),$$ it is shown that a set of three transformations can be introduced which completely define the Sequential Probability Ratio Test for testing a hypothesis $H_0$ against $H_1$. When the observer specifies the threshold parameters $\theta_0$ and $\theta_1$ corresponding to the hypotheses $H_0$ and $H_1$ and the strength $\alpha, \beta$ of the test, he specifies the three transformations and hence the Sequential Test. However, there is an infinity of sets of parameter points $(\theta_0, \theta_1, \alpha, \beta)$ which satisfy the same transformations and hence define the same Sequential Test. The Operating Characteristic Function and the Average Sample Number Function are derived in terms of these transformations.

The problem of obtaining maximum likelihood estimates for the parameters involved in a stationary single-channel, Markovian queuing process is considered. A method of taking observations is presented which simplifies this problem to that of determining a root of a certain quadratic equation. A useful and even simpler rational approximation is also studied.

Three tests were considered by Severo and Olds [1] for testing an hypothesis on a mean of a logarithmic-normal distribution with known variance. The purpose of this note is to discuss the asymptotic behavior of these tests for large sample size.

Let $r$ be the sample serial correlation coefficient computed from a sample of size $N$ drawn from a serially correlated process with parameter $\rho$. It is shown that the statistic $$t = \frac{(r - \rho) \sqrt{N + 1}}{\sqrt{1 - r^2}}$$ is approximately distributed as Student's $t$ with $N + 1$ degrees of freedom.

The conditional Monte Carlo technique advanced by Tukey et al. [1, 2] has been explained in analytic terms by Hammersley [3]. This note offers an alternative explanation, wherein the group-theoretic aspect of the problem plays the dominant role. The method is illustrated on an example simpler than that treated in [1, 2].