In this paper linear restrictions on regression coefficients are studied. Let the $p \times q_2$ matrix of coefficients of regression of the $p$ dependent variates on $q_2$ of the independent variates be $\mathbf{\bar B}_2$. Maximum likelihood estimates of an $m \times p$ matrix $\Gamma$ satisfying $\Gamma'\mathbf{\bar B}_2 = 0$ and certain other conditions are found under the assumption that the rank of $\mathbf{\bar B}_2$ is $p - m$ and the dependent variates are normally distributed (Section 2). Confidence regions for $\Gamma$ under various conditions are obtained (Section 5). The likelihood ratio test of the hypothesis that the rank of $\mathbf{\bar B}_2$ is a given number is obtained (Section 3). A test of the hypothesis that $\Gamma$ is a certain matrix is given (Section 4). These results are applied to the "$q$-sample problem" (Section 7) and are extended for certain econometric models (Section 6).

The first part of this paper considers two methods of estimating the linear structural relation between two variables both of which are subject to "error"; the second part of the paper comments on a recently advanced procedure for constructing the confidence region for the slope of the structural relation. In 1940 Wald [1] initiated a certain procedure for estimating the linear structural relation between two variables both of which are subject to "error." Wald's idea was extended by Nair and Banerjee [2] and later by Bartlett [3]. These procedures require some knowledge about the values of certain non-observable variables. When this knowledge is not available there is a temptation to substitute information derived from observations. One such method was considered by Wald who found sufficient conditions for the consistency of the resulting estimate. The purpose of the first part of the present paper is to find the necessary and sufficient conditions for two procedures with reference to a slightly more general case, namely, when the "errors" in the two observable variables may be correlated. The results obtained indicate that the two procedures, applied in the case of no additional knowledge about the values of the non-observable variables, will lead to consistent estimates of the slope of the structural relation in very exceptional cases only. In 1949 Hemelrijk [4] described a novel procedure for constructing the confidence region of the slope of the linear structural relation in the case when the non-observable variables have unknown fixed values and the observations are made with "error" which has the same probability distribution at each point. The present paper considers this same procedure when there is no information about the fixed non-observable variables and also when these variables are random variables, and shows that the probability that the confidence region covers the true slope is the same as before but that the probability of covering any other slope is now the same as this probability of covering the true slope.

Some methods for testing independence between the components of a random vector are discussed. The basic principle in the construction of the tests is dichotomization of each component variable. The distributions are obtained under randomization. Other applications of the tests are mentioned (Section 2). Certain limiting distributions are derived (Section 4). The exact distribution of the test statistic in a special case is tabulated (Section 5). A brief study of an alternative test is made (Section 6).

In 1943 Wald [2] gave a method for constructing tolerance regions in the multivariate case. Tukey generalized Wald's procedure in [4] and the results were interpreted for discontinuous distributions in [5] and [6]. This paper presents a further generalization of the method so that statistically equivalent blocks can be determined sequentially; the particular function used to cut off a block may depend on the shape or structure of previously selected blocks. The results are also interpreted for the case of discontinuous distributions. Possible advantages of applying the method are discussed.

The method considered here for investigating the effect of non-normality and heterogeneity of variance on tests of the general linear hypothesis is based on finding the cumulants of a linear function of the two sums of squares used in the usual $F$-test.

For any chance variable $x = (x_1,\cdots,x_N)$ having known distribution, the translation parameter estimation problem is to estimate an unknown constant $h$, having observed $y = (x_1 + h,\cdots,x_N + h)$. Extending the work of Pitman [2], Girshick and Savage [1] have, for any loss function depending only on the error of estimate, described an estimate whose risk is a constant $R$ independent of $h$, and have shown that under certain hypotheses their estimate is minimax. We investigate whether the Girshick-Savage estimate is admissible, i.e., whether it is impossible to find an estimate with risk $R(h) \leq R$ for all $h$ and actual inequality for some $h$. We consider only bounded discrete variables $x$, and show that, if all values of $x$ have all integer coordinates and if the loss $f(d)$ from an error $d$ is, for instance, strictly convex and assumes its minimum value, the Girshick-Savage estimate is admissible. Two examples in which the Girshick-Savage estimate is not admissible are given.

Let $M(x)$ denote the expected value at level $x$ of the response to a certain experiment. $M(x)$ is assumed to be a monotone function of $x$ but is unknown to the experimenter, and it is desired to find the solution $x = \theta$ of the equation $M(x) = \alpha$, where $\alpha$ is a given constant. We give a method for making successive experiments at levels $x_1,x_2,\cdots$ in such a way that $x_n$ will tend to $\theta$ in probability.

In addition to being easily applied and reasonably efficient for small samples, the equal-tail sign procedure for testing hypotheses about, or setting confidence intervals for, the population median is valid under very general conditions. (For brevity, the equal-tail sign procedure will be referred to as Procedure E.) Rarely, if ever, however, are these conditions exactly satisfied in practice. Thus the actual significance level or confidence coefficient for Procedure E is only an approximation to the standard value (which holds when the conditions are satisfied). Undoubtedly the equal-tail sign procedure is used in many cases when these conditions are only roughly approximated. The purpose of this paper is to investigate under what conditions Procedure E has significance levels and confidence coefficients which are satisfactory approximations to the standard values. It is found that the approximation is reasonably good for a wide variety of situations if the number of observations is not large. Thus, as far as errors of Type I are concerned, Procedure E is a sufficiently close approximation for many practical cases. This significance level stability, combined with its other favorable properties, suggests that the equal-tail sign procedure be seriously considered for application when an inference is to be made from a small number of observations to the population median.

The asymptotic distribution of the range and normalized range of the sum of $n$ independent variables is derived using the theory of Brownian motion.

Certain inequalities are obtained for the moments of unimodal distributions.

Bhattacharyya [1] has considered recently the following problem in statistical estimation. Let $X_1,X_2,\cdots,X_n$ be $n$ stochastic variables distributed according to the probability law $f(x_1,x_2,\cdots,x_n; \theta) dx_1dx_2\cdots dx_n$, where $\theta$ is the unknown parameter. Consider the class of all functions $T(x_1,x_2,\cdots,x_n)$ of the stochastic variables such that the expectation of each function in this class is equal to a preassigned function $\tau(\theta)$. Usually $\tau(\theta)$ admits of more than one unbiased estimate, and the problem posed by various authors is to obtain a lower bound of the variances of all such estimates, this lower bound to be independent of the estimates themselves but depending on $\tau(\theta)$ and the distribution function of the $n$ stochastic variables. Under certain regularity conditions Bhattacharyya obtained a lower bound of the above type which is never lower than the one obtained earlier independently by Cramer [2] and Rao [3], although the conditions assumed by Bhattacharyya are more restrictive than those assumed by the latter authors. Recently E. W. Barankin in a remarkable paper [4] has developed a procedure which yields the class of lower bounds of unbiased estimates having minimum $s$th absolute central moment $(s > 1)$ at a preassigned parameter value $\theta_0$. In this note we are concerned with the attainment of a lower bound obtained first by Bhattacharyya. Bhattacharyya discusses the case in which his lower bound is attained and derives some interesting properties of the distribution of such a statistic (which might be called a generalized efficient statistic). The purpose of this note is to prove that in the case in which the variables $X_1,X_2,\cdots,X_n$ are independently and identically distributed with a common distribution function $F(x; \theta)$ depending upon a single unknown parameter, one obtains the following result: under the regularity conditions assumed by Bhattacharyya in which the parameter $\theta$ may assume values in an interval of the real axis, and with an additional slight restriction on the cumulative distribution function $F(x; \theta)$, no generalized efficient statistic exists which is constructed by use of both the first and second derivatives of the likelihood function with respect to the parameter. It follows that if an efficient estimate (in the sense originally defined by Fisher [5]) for the single unknown parameter does not exist, then no distribution $F(x; \theta)$ exists possessing a uniformly minimum variance unbiased estimate of $\tau(\theta)$ which is constructed by using a linear combination of the first and second partial logarithmic derivatives of the likelihood function. This result for the case involving a single unknown parameter is peculiarly of interest in view of the fact that Seth [6] has given an example in which the above construction is possible if the distribution involves two unknown parameters.

For testing the hypothesis that successive members of a series of observations are independent J. von Neumann [5] (see also B. I. Hart [4]) and R. L. Anderson [1] have proposed test statistics and tabulated their significance points. von Neumann's criterion seems well designed to detect deviations from the null hypothesis which might be encountered in practice but its exact distribution is unknown. On the other hand Anderson's statistic, while it has a known distribution, is based on a circular conception of the population which is rarely plausible in practice. In the present note certain noncircular statistics are proposed for which exact distributions can be obtained from Anderson's results. The statistics are derived from the usual noncircular statistics by sacrificing a small amount of relevant information. Their application is noted to certain regression problems for which no satisfactory tests are at present available. Finally, some general remarks are made about the choice of best statistics for the problems discussed.

Let $n$ variates possessing finite first and second moments be partitioned into $k$ sets. A system of equations is developed for which some solution consists of $k$ sets of coefficients which combine the $k$ sets of variates into $k$ variates possessing minimum generalized variance.

In a test of an hypothesis one may regard a sample in the critical region as evidence that the hypothesis is false. Let us assume that for some reason it is desired to increase the critical size of the test, i.e., to make rejection of the hypothesis more probable. Then one may expect that an observation which led to rejection in the first test should still lead to rejection in the new test. In other words, one should expect $W_\alpha \supset W_{\alpha'}$ if $\alpha > \alpha'$, where $W_\alpha$ is the critical region for the test of size $\alpha$. An example is given where regions of type $A$ are uniquely specified except for sets of measure zero, but fail to have this property.