The asymptotic distribution of a class of statistics, which has been called serial statistics, has been obtained, for permutations of the observed sample values. Specific instances of the use of such statistics, for the test of randomness of a sequence, have been given and the large sample power functions have been considered, when the alternative is a Markov process.

A single-sample multiple decision procedure for ranking variances of normal populations is described. Exact small-sample methods and a large-sample method are given for computing the sample sizes necessary to guarantee a preassigned probability of a correct ranking under specified conditions on certain variance ratios. Some tables computed by these methods are provided.

This is the first of two papers describing a study of the effect of departures from assumptions, other than normality, on the null-distribution of the $F$-statistic in the analysis of variance. In this paper, certain theorems required in the study and concerning the distribution of quadratic forms in multi-normally distributed variables are first enunciated and simple approximations tested numerically. The results are then applied to determine the effect of group-to-group inequality of variance in the one-way classification. It appears that if the groups are equal, moderate inequality of variance does not seriously affect the test. However, with unequal groups, much larger discrepancies appear. In a second paper, similar methods are used to determine the effect of inequality of variance and serial correlation between errors in the two-way classification.

The purpose of this paper is to compare for various two-parameter distributions, of the form $f\{(x - \mu)/\sigma\}/\sigma,$ the estimates of the parameters obtained by applying the method of least squares to the observations, after these have been arranged in order of magnitude. Estimates obtained by this process we shall call "ordered least-squares estimates." Such estimates are unbiased and have minimal variance among all unbiased estimates which are linear in the ordered observations. This estimation process has been previously discussed by Godwin [1] and [2] and Lloyd [3]. In the present paper, ordered estimates are obtained explicitly for a class of two-parameter distributions having the above form. This class contains the rectangular and the right triangular distributions as special cases. It also reduces to the exponential distribution as a limiting case. Other special cases of this class of distributions have also been previously discussed by Craig [4]. Further, a general property of ordered least-squares estimates of the parameter $\lambda$ in distributions of the type $f(x/\lambda)/\lambda$ is discussed. As a result it is shown that the ordered least-squares estimate of the scale parameter in the Pearson Type III distribution is identical with the maximum likelihood estimate.

The aim of this paper is: (i) to find the best linear estimates of the means and standard deviations of the rectangular, triangular, exponential and double exponential populations; (ii) to compare the efficiencies of these estimates with some other estimates for small samples; (iii) to discuss the variation of coefficients in the best linear estimates as the population varies.

This paper is concerned with a matrix method of deriving the sampling distributions of a large class of statistics directly from the probability law for random samples from a multivariate normal population, that is without assuming the Wishart distribution or the distribution of rectangular coordinates. Two techniques are proposed for evaluating the Jacobians of certain transformations, one based on a theorem on Jacobians [1], and the second based on the introduction of pseudo or extra variables. This matrix approach has a geometrical analog developed in part by one of the authors [2]. Section 3 is concerned with a discussion of these two techniques; in Section 4, the former is applied to obtain the joint distribution of the rectangular coordinates [3], and in Section 5, the second method is applied to obtain the joint distribution of the roots of a determinantal equation [4], [5], [6], and [7].

The connection between the theory of random walks and Wald's theory [10], [11] of sequential probability ratio tests of hypotheses has been remarked by several authors. In particular, Kemperman [5] has exploited that connection to obtain integral equations for the determination of the decision probabilities and the expected sample size of a Wald sequential test. It is the purpose of the present paper (1) to generalize Kemperman's integral equations to apply to a fairly extensive class of sequential multiple decision problems, and (2) to indicate methods of obtaining practical results from such integral equations. Part I of the paper is purely theoretical. It presents the integral equations already mentioned and generalizes a method of obtaining upper and lower bounds for their solutions that seems to have been first published by Kemperman [5] and Snyder [7] simultaneously. In Part II the possibilities for application of the general theory are illustrated by a discussion of Wald's sequential tests for simple alternatives on the parameter of a distribution, under the hypothesis that a sufficient statistic for that parameter exists. In particular, it is shown that the Kemperman-Snyder method for obtaining bounds for the solutions of the integral equations may be used to obtain substantial improvements over the bounds given by Wald for the operating characteristics of the test for simple alternatives on the mean of a normal distribution. Methods of numerical analysis are indicated that might be useful in a well-equipped computing laboratory for further improvement of the bounds. It is clear from the results obtained here that the methods used, coupled with extensive numerical work, should yield definitive improvements over Wald's approximate methods for setting the decision boundaries and estimating the sample size moments for sequential tests. It is hoped that the decision rule adopted in Part I is sufficiently general that the theory will provide a useful tool in the design and study of multiple decision problems.

The primary purpose of this paper is to discuss certain sequential procedures for discriminating between two values for the ratio of variance components in a simple one-way classification. In order to clarify the presentation of the three procedures discussed, they will first be applied (in Section 2) to the problem of discrimination between two values for the ratio of variances in two distinct normal populations. The arguments in Section 3 closely parallel those in Section 2, and the form of the two sections has been made as similar as possible, to emphasize this parallelism. Section 4 contains some formulae used in calculating approximate average sampling numbers presented in the earlier sections.

In this paper quadratic estimates of variance components are considered. For the general balanced nested classification with no specific distributions assumed, it is shown that the quadratic estimate which is unbiased and which has minimum variance is given by the analysis of variance method of estimating the variance components.

A life test on $N$ items is considered in which the common underlying distribution of the length of life of a single item is given by the density \begin{equation*}\tag{1} p(x; \theta, A) = \begin{cases}\frac{1}{\theta} e^{-(x-A)/\theta},\quad\text{for} x \geqq A \\ 0,\quad\text{otherwise}\end{cases}\end{equation*} where $\theta > 0$ is unknown but is the same for all items and $A \geqq 0.$ Several lemmas are given concerning the first $r$ out of $n$ observations when the underlying p.d.f. is given by (1). These results are then used to estimate $\theta$ when the $N$ items are divided into $k$ sets $S_j$ (each containing $n_j > 0,$ items, $\sum^k_{j=1} n_j = N)$ and each set $S_j$ is observed only until the first $r_j$ failures occur $(0 < r_j \leqq n_j).$ The constants $r_j$ and $n_j$ are fixed and preassigned. Three different cases are considered: 1. The $n_j$ items in each set $S_j$ have a common known $A_j (j = 1, 2, \cdots, k).$ 2. All $N$ items have a common unknown $A.$ 3. The $n_j$ items in each set $S_j$ have a common unknown $A_j (j = 1, 2, \cdots, k).$ The results for these three cases are such that the results for any intermediate situation (i.e. some $A_j$ values known, the others unknown) can be written down at will. The particular case $k = 1$ and $A = 0$ is treated in [2]. The constant $A$ in (1) can be interpreted in two different ways: (i) $A$ is the minimum life, that is life is measured from the beginning of time, which is taken as zero. (ii) $A$ is the "time of birth", that is life is measured from time $A$. Under interpretation (ii) the parameter $\theta,$ which we are trying to estimate, represents the expected length of life.

Let $H(y\mid x)$ be a family of distribution functions depending upon a real parameter $x,$ and let $M(x) = \int^\infty_{-\infty} y dH(y \mid x)$ be the corresponding regression function. It is assumed $M(x)$ is unknown to the experimenter, who is, however, allowed to take observations on $H(y\mid x)$ for any value $x.$ Robbins and Monro [1] give a method for defining successively a sequence $\{x_n\}$ such that $x_n$ converges to $\theta$ in probability, where $\theta$ is a root of the equation $M(x) = \alpha$ and $\alpha$ is a given number. Wolfowitz [2] generalizes these results, and Kiefer and Wolfowitz [3], solve a similar problem in the case when $M(x)$ has a maximum at $x = \theta.$ Using a lemma due to Loeve [4], we show that in both cases $x_n$ converges to $\theta$ with probability one, under weaker conditions than those imposed in [2] and [3]. Further we solve a similar problem in the case when $M(x)$ is the median of $H(y \mid x).$

The following theorem is proved. Let $X_1, X_2, \cdots, X_n$ be $n$ independently (but not necessarily identically) distributed random variables, and assume that the $n$th moment of each $X_i(i = 1, 2, \cdots, n)$ exists. The necessary and sufficient conditions for the existence of two statistically independent linear forms $Y_1 = \sum^n_{s=1} a_sX_s$ and $Y_2 = \sum^n_{s=1}b_sX_s$ are: (A) Each random variable which has a nonzero coefficient in both forms is normally distributed. $(B) \sum^n_{s=1}a_sb_s\sigma^2_s = 0$. Here $\sigma^2_s$ denotes the variance of $X_s (s = 1, 2, \cdots, n)$. For $n = 2$ and $a_1 = b_1 = a_2 = 1, b_2 = -1$ this reduces to a theorem of S. Bernstein [1]. Bernstein's paper was not accessible to the authors, whose knowledge of his result was derived from a statement of S. Bernstein's theorem contained in a paper by M. Frechet [3]. A more general result, not assuming the existence of moments was obtained earlier by M. Kac [4]. A related theorem, assuming equidistribution of the $X_i (i = 1, 2, \cdots n)$ is stated without proof in a recent paper by Yu. V. Linnik [5].

For any sequence $x_1, x_2, \cdots$ of chance variables satisfying $| x_n | \leqq 1$ and $E(x_n\| x_1, \cdots, x_{n-1} \leqq -u(\max | x_n\| x_1, \cdots, x_{n-1})$, where $u$ is a fixed constant, $0 < u < 1$, and for any positive number $t$, $\mathrm{Pr} \{ \underset{n}\sup (x_1 + \cdots + x_n) \geqq t\} \leqq \big(\frac{1 - u}{1 + u}\big)^t.$ Equality holds for integral $t$ when $x_1, x_2, \cdots$ are independent with $\mathrm{Pr} \{x_n = 1\} = (1 - u)/2, \quad \mathrm{Pr} \{x_n = -1\} = (1 + u)/2.$ This has a simple interpretation in terms of gambling systems, and yields a new proof of Levy's extension of the strong law of large numbers to dependent variables [2], with an improved estimate for the rate of convergence.

A new concept of unbiasedness (density unbiasedness) for point estimates is introduced and the "best" density unbiased point estimate for the mean of any normal distribution is proved to be the ordinary sample mean $\bar{x} = \sum^n_{i = 1} x_i/n$. Under certain conditions on the form of the characteristic function of a family of probability density functions involving an unknown location parameter, $\bar{x}$ is shown to be a density unbiased point estimate of the location parameter.

The object of this paper is to present certain important properties of the Gamma distribution and the two kinds of Beta distributions, and to indicate certain useful applications of these two to sampling problems. The distribution of the Studentised $D^2$-statistic under the null hypothesis is obtained in two different ways.

In this note, we prove that, for all (finite) values of $h$, \begin{equation*}\tag{1} \psi(h) = \frac{m_2}{m^2_1} = \frac{1 - h(Z - h)}{(Z - h)^2},\end{equation*} is monotonic increasing, that \begin{equation*}\tag{2} 2m^2_1 - m_2 > 0,\end{equation*} and that \begin{equation*}\tag{3} 1 < \psi(h) < 2,\end{equation*} where $Z$ is the reciprocal of Mill's ratio, \begin{equation*}\tag{4} Z(h) = e^{-h^2/2} \big/ \int^\infty_h e^{-t^2/2} dt,\end{equation*} and where $m_1$ and $m_2$ are respectively the first and second moments of a singly truncated normal distribution about the point of truncation. The function $\psi(h)$ arises in connection with maximum likelihood estimation of population parameters from singly truncated normal samples (cf. for example [1] and references cited therein). The inequality (2) arises in connection with three-moment estimates based on samples of the same type (cf. [2] and [3]).