Let $Z_n$ denote the system of $8n$ independent pairs of measurements $(X_{ik}, Y_{ik}),$ for $4i = 1, 2, \cdots, n$ and $k = 1, 2, \cdots, 8,$ of two nonobservable random variables $\xi_{ik}$ and $\eta_{ik},$ known to satisfy a linear relation of the form $\xi_{ik} \cos \theta^\ast + \eta_{ik} \sin \theta^\ast - p = 0,$ where $p$ is an arbitrary real number and $\theta^\ast$ may have any value between the limits $-\frac{1}{2}\pi < \theta^\ast \leqq \frac{1}{2}\pi$. The purpose of the paper is to construct a class of estimates $T_n(Z_n)$ of the parameter $\theta$ defined as follows: when $\theta^\ast = \frac{1}{2}\pi$ then $\theta = 0$; otherwise $\theta \equiv \theta^\ast$. Each estimate $T_n(Z_n)$ of the class considered converges in probability to $\theta$ as $n \rightarrow \infty$ under the following conditions: (i) except when $\theta = 0$, the variables $\xi_{ik}$ are nonnormal; (ii) any nonnormal components of the errors of measurements, $X_{ik} - \xi_{ik}$ and $Y_{ik} - \eta_{ik},$ are mutually independent, independent of $\xi_{ik}$ and of the normal components of these errors; (iii) the normal components of the errors may be correlated but as a pair are independent of $\xi_{ik}$.

Hotelling's [1] theoretical findings in mathematical economics on the rational behavior of buyers in maximizing their net profit indicate that the matrix of the first partial derivatives of a set of related demand functions would be symmetric and negative definite. It is the object of this paper to determine whether the assumption of symmetry will be tenable in the light of the particular set of observations. The study of test functions for the property of definiteness as a whole will form the subject of a forthcoming paper. The present investigation assumes that the demand functions are regression functions and, therefore, results obtained in this paper do not cover all types of demand functions. The test function $U$ proposed here for the hypothesis of symmetry is invariant under all contragredient transformations. The distribution of $U$ depends on unknown nuisance parameters. The likelihood ratio under the hypothesis of symmetry leads to a multilateral matric equation which represents $\frac{1}{2} p(p + 1)$ equations of the third degree in $\frac{1}{2} p(p + 1)$ unknown regression coefficients for the $p$-variate case. It has not been possible to establish the existence of a nontrivial solution of this equation, and it is, therefore, not being given here.

The upper and lower bounds for the expectation, the coefficient of variation, and the variance of the largest member of a sample from a symmetric population are discussed. The upper bound for the expectation (Table 1, Fig. 1), the lower bound for the C.V. (Table 2, Fig. 4) and the lower bound for the variance (Fig. 7) are actually achieved for the corresponding particular population distributions (Figs. 2, 3, 5, 6, equation (5.1)). The rest of the bounds are not actually achieved but approached as the limits, for example, for the three-point distribution (Section 3) by letting $p$ tend to zero.

A method of fitting polynomials to equally spaced data is developed which is more rapid than the method of least squares. The orthogonal polynomial $T_j(x)$ of the least squares method is replaced by a step function $w_j(x),$ and this greatly reduces the number of multiplications. An efficiency of about 90 per cent is obtained for the estimates of the coefficients and fitted values.

Let $(Y_{n1}, \cdots, Y_{nn})$ be a random vector which takes on the $n!$ permutations of $(1, \cdots, n)$ with equal probabilities. Let $c_n(i, j), i,j = 1, \cdots, n,$ be $n^2$ real numbers. Sufficient conditions for the asymptotic normality of $S_n = \sum^n_{i=1} c_n(i, Y_{ni})$ are given (Theorem 3). For the special case $c_n(i,j) = a_n(i)b_n(j)$ a stronger version of a theorem of Wald, Wolfowitz and Noether is obtained (Theorem 4). A condition of Noether is simplified (Theorem 1).

In this paper we prove that certain estimators and tests of significance used in regression analysis when observations are independent are equally valid in the presence of intra-class correlation. An application of this result is presented for the situation in which several replications of the correlated set of observations are available. As a special case of this application, it is shown that the usual test of ``column effects'' in the analysis of variance for a two-way classification remains valid when rows are independent and columns are uniformly correlated. This latter fact is also pointed out in [3].

Following the essential steps of the proof of the Cramer-Rao inequality [1, 2] but avoiding the need to transform coordinates or to differentiate under integral signs, a lower bound for the variance of estimators is obtained which is (a) free from regularity assumptions and (b) at least equal to and in some cases greater than that given by the Cramer-Rao inequality. The inequality of this paper might also be obtained from Barankin's general result [3]. Only the simplest case--that of unbiased estimation of a single real parameter--is considered here but the same idea can be applied to more general problems of estimation.

Let $F(x)$ be the continuous distribution function of a random variable $X,$ and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$. It is well known that the probability $P_n(\epsilon)$ of $F(x)$ being everywhere majorized by $F_n(x) + \epsilon$ is independent of $F(x)$. The present paper contains the derivation of an explicit expression for $P_n(\epsilon)$, and a tabulation of the 10%, 5%, 1%, and 0.1% points of $P_n(\epsilon)$ for $n =$ 5, 8, 10, 20, 40, 50. For $n =$ 50 these values agree closely with those obtained from an asymptotic expression due to N. Smirnov.

For samples of any given size $N \geq 2$ from a normal population, Wilks [1] has shown how to choose the parameter $\lambda_p$ so that the expected coverage of the interval $\bar x \pm \lambda_ps$ will be $1 - p$. The present paper treats the choice of the minimal sample size $N$ necessary to effect a certain type of statistical control on the fluctuation of that coverage about its expected value; a brief table of such minimal sample sizes is given.