Let $X_1, \cdot, X_n$ be $n$ independent random vectors, $X_\nu = (X^{(1)}_\nu, \cdots, X^{(r)}_\nu),$ and $\Phi(x_1, \cdots, x_m)$ a function of $m(\leq n)$ vectors $x_\nu = (x^{(1)}_\nu , \cdots, x^{(r)}_\nu)$. A statistic of the form $U = \sum"\Phi(X_{\alpha 1}, \cdots, X_{\alpha_m})/n(n - 1) \cdots (n - m + 1),$ where the sum $\sum"$ is extended over all permutations $(\alpha_1, \cdots, \alpha_m)$ of $m$ different integers, $1 \leq \alpha_i \leq n$, is called a $U$-statistic. If $X_1, \cdots, X_n$ have the same (cumulative) distribution function (d.f.) $F(x), U$ is an unbiased estimate of the population characteristic $\theta(F) = \int \cdots \int\Phi(x_1, \cdots, x_m) dF(x_1) \cdots dF(x_m). \theta(F)$ is called a regular functional of the d.f. $F(x)$. Certain optimal properties of $U$-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4). The variance of a $U$-statistic as a function of the sample size $n$ and of certain population characteristics is studied in Section 5. It is shown that if $X_1, \cdots, X_n$ have the same distribution and $\Phi(x_1, \cdots, x_m)$ is independent of $n$, the d.f. of $\sqrt n(U - \theta)$ tends to a normal d.f. as $n \rightarrow \infty$ under the sole condition of the existence of $E\Phi^2(X_1, \cdots, X_m)$. Similar results hold for the joint distribution of several $U$-statistics (Theorems 7.1 and 7.2), for statistics $U'$ which, in a certain sense, are asymptotically equivalent to $U$ (Theorems 7.3 and 7.4), for certain functions of statistics $U$ or $U'$ (Theorem 7.5) and, under certain additional assumptions, for the case of the $X_\nu$'s having different distributions (Theorems 8.1 and 8.2). Results of a similar character, though under different assumptions, are contained in a recent paper by von Mises [18] (cf. Section 7). Examples of statistics of the form $U$ or $U'$ are the moments, Fisher's $k$-statistics, Gini's mean difference, and several rank correlation statistics such as Spearman's rank correlation and the difference sign correlation (cf. Section 9). Asymptotic power functions for the non-parametric tests of independence based on these rank statistics are obtained. They show that these tests are not unbiased in the limit (Section 9f). The asymptotic distribution of the coefficient of partial difference sign correlation which has been suggested by Kendall also is obtained (Section 9h).

Let $S_0$ be any sequential probability ratio test for deciding between two simple alternatives $H_0$ and $H_1$, and $S_1$ another test for the same purpose. We define $(i, j = 0, 1):$ $\alpha_i(S_j) =$ probability, under $S_j$, of rejecting $H_i$ when it is true; $E_i^j (n) =$ expected number of observations to reach a decision under test $S_j$ when the hypothesis $H_i$ is true. (It is assumed that $E^1_i (n)$ exists.) In this paper it is proved that, if $\alpha_i(S_1) \leq \alpha_i(S_0)\quad(i = 0,1)$, it follows that $E_i^0 (n) \leq E_i^1 (n)\quad(i = 0, 1)$. This means that of all tests with the same power the sequential probability ratio test requires on the average fewest observations. This result had been conjectured earlier ([1], [2]).

The exact distribution of a root of a determinantal equation when the roots are arranged in a monotonic order was obtained by S. N. Roy [3] in 1943. A different method for deriving the distribution of any one of these roots has been described by the author in [2]. In the present paper the limiting forms of these distributions are obtained. This paper gives a method by which the limiting distributions can be obtained without undergoing an inordinate amount of mathematical labor.

It is shown that if, in the analysis of variance, the experiments are not in a state of statistical control due to variations in the true means, then the test will have a downward bias. The power function of the analysis of variance test is obtained when this downward bias is present.

Mixtures of measures or distributions occur frequently in the theory and applications of probability and statistics. In the simplest case it may, for example, be reasonable to assume that one is dealing with the mixture in given proportions of a finite number of normal populations with different means or variances. The mixture parameter may also be denumerably infinite, as in the theory of sums of a random number of random variables, or continuous, as in the compound Poisson distribution. The operation of Lebesgue-Stieltjes integration, $\int f(x) d\mu ,$ is linear with respect to both integrand $f(x)$ and measure $\mu$. The first type of linearity has as its continuous analog the theorem of Fubini on interchange of order of integration; the second type of linearity has a corresponding continuous analog which is of importance whenever one deals with mixtures of measures or distributions, and which forms the subject of the present paper. Other treatments of the same subject have been given ([1], [2]; see also [3], [4]) but it is hoped that the discussion given here will be useful to the mathematical statistician. A general measure theoretic form of the fundamental theorem is given in Section 2, and in Section 3 the theorem is formulated in terms of finite dimensional spaces and distribution functions. The operation of convolution as an example of mixture is treated briefly in Section 4, while Section 5 is devoted to random sampling from a mixed population. We shall refer to Theory of the Integral by S. Saks (second edition, Warszawa, 1937) as [S], and the Mathematical Methods of Statistics by H. Cramer (Princeton, 1946) as [C].

It is well known that the Fourier transform is a powerful analytical tool in studying the distribution of sums of independent random variables. In this paper it is pointed out that the Mellin transform is a natural analytical tool to use in studying the distribution of products and quotients of independent random variables. Formulae are given for determining the probability density functions of the product and the quotient $\frac{\xi}{\eta}$, where $\xi$ and $\eta$ are independent positive random variables with p.d.f.'s $f(x)$ and $g(y)$, in terms of the Mellin transforms $F(s) = \int_0^\infty f(x) x^{s-1} dx$ and $G(s) = \int_0^\infty g(y)y^{s-1} dy$. An extension of the transform technique to random variables which are not everywhere positive is given. A number of examples including Student's $t$-distribution and Snedecor's $F$-distribution are worked out by the technique of this paper.

This paper deals with the problem of bivariate regression where both variates are random variables having a finite number of means distributed along a straight line. A regression statistic is derived which is independent of change in scale so that a prior knowledge of the frequency distribution parameters is not required in order to obtain a unique estimate. The statistic is shown to be consistent. The efficiency of the estimate is discussed and its asymptotic distribution is derived for the case when the random variables are normally distributed. A numerical example is presented which compares the performance of the statistic of this paper with that of other commonly used statistics. In the example it is found that the method of estimation proposed in this paper is more efficient.

Let $t$ be the true value of what is being measured and suppose that the error of observation is a symmetric normal distribution of standard deviation $\sigma$. The "rounding-off" error due to the reading of measurements to the nearest unit has a distribution and an expected value depending on $t$ and $\sigma$. It is shown that, for a fixed $\sigma > 0$, the expected value of the decimal correction, $r(t; \sigma)$, is an analytic function of $t$ which is odd, of period 1, positive for $0 < t < \frac{1}{2}$, and has a convex arch as its graph on $0 \leqq t \leqq \frac{1}{2}$. Furthermore, if $0 < t < \frac{1}{2}$, both $r(t; \sigma)$ and its maximum value, $\operatorname{Max}_t r(t; \sigma)$, are decreasing functions of $\sigma$.

Consider a random sample of $N$ observations $x_1, x_2, \cdots, x_N,$ from a universe of mean $\mu$ and variance $\sigma^2$. Let $m$ and $s^2$ be the sample mean and variance respectively: \begin{equation*}\tag{1} m = \frac{1}{N} \sum^N_{i=1} x_i,\quad s^2 = \frac{1}{N} \sum^N_{i=1} (x_i - m)^2.\end{equation*} It is shown that the following conservative confidence interval holds for $\mu$: \begin{equation*}\tag{2} \operatorname{Prob}\{ (m - \mu)^2 \leqq s^2/(N - 1) + \lambda\sigma^2\sqrt{2/N(N - 1)}\} > 1 - \lambda^{-2},\end{equation*} where $\lambda$ is any positive constant. Inequality (2) also holds if, in the braces, $\lambda$ is replaced by $\sqrt{\lambda^2 - 1}$, with $\lambda \geqq 1$. Inequality (2) is much more efficient on the average than Tchebychef's inequality for the mean, namely, \begin{equation*}\tag{3} \operatorname{Prob} \{(m - \mu)^2 \leqq \lambda^2\sigma^2/N\} > 1 - \lambda^{-2},\end{equation*} yet (2) and (3) are both distribution-free, requiring only knowledge about $\sigma^2$. At the $1 - \lambda^{-2} = .99$ level of confidence, the expected value of the right member in the braces of (2) is only about $1/6$ the corresponding member of (3); at the .999 level of confidence the ratio is about $1/20$. A more general inequality than (2) is developed, also involving only the single parameter $\sigma^2$.*

In this note we deduce several properties of the compound and generalized Poisson distributions; in particular their closure and divisibility properties. An infinite class of functions whose members are both compound and generalized Poisson distributions is exhibited, and several of the distributions of Neyman, Polya, etc. are identified. The present note stems from a paper by Feller [2].