In this paper statistical test criteria are developed for testing equality of means, equality of variances and equality of covariances in a normal multivariate population of $k$ variables on the basis of a sample. More specifically, three statistical hypotheses are considered: (i) $H_{mvc}$, the hypothesis that the means are equal, the variances are equal, and covariances are equal, (ii) $H_{vc}$, the hypothesis that variances are equal and covariances are equal, irrespective of the values of the means, and (iii) $H_m$, the hypothesis of equal means, assuming variances are equal and covariances are equal. Test criteria $L_{mvc}$, $L_{vc}$, and $L_m$ are developed by the Neyman-Pearson method of likelihood ratios for testing $H_{mvc}$, $H_{vc}$ and $H_m$ respectively. The exact moments of each of the three test criteria when the three corresponding hypotheses are true are determined for any number $k$ of variables and for any size, $n$, of the sample for which the distributions exist. The exact distributions of $L_{mvc}$ and $L_{vc}$ are determined for $k = 2$ and $k = 3$, and the exact distribution of $L_m$ is found for any $k$; these are all beta (Pearson Type I) distributions. Tables of 5% and 1% points of $L_{mvc}$, $L_{vc}$ and $L_m$, based on Thompson's tables of percentage points of the Incomplete Beta Function, are given for certain values of $k$ and $n$ (Tables I and II). Also tables of values of approximate 5% and 1% points of $-n \ln L_{mvc}, -n \ln L_{vc}$ and $-n(k - 1) \ln L_m$ for large values of $n$ are given (Table III), based on the fact that these three quantities are approximately distributed according to chi-square laws for large values of $n$ with $\frac{1}{2}k(k + 3) - 3, \frac{1}{2}k(k + 1) - 2$, and $k - 1$ degrees of freedom respectively. A table (Table IV) is given which shows how accurate the resulting approximate 5% and 1% points of $L_{mvc}, L_c$ and $L_m$ are. The paper is written in two parts. In Part I the problem of testing the three hypotheses is discussed and the mathematical results are presented together with an illustrative example. Part II is given for the reader who wishes to study the mathematical derivation of the results.

This is a continuation of a paper Part I of which was published in the June, 1946 issue of the Annals of Mathematical Statistics. The present paper is divided into two parts, Parts II and III, which are summarized as follows: Part II. The Exact Power Curve and the Distribution of n for Sequential Tests Where z Takes on a Finite Number of Integral Values. Consider a sequential test defined by a decision function $Z_n = \sum^n_{\alpha = 1} z_\alpha$ with boundaries $-b$ and $a$ where $a$ and $b$ are positive integers and $z_\alpha$ is the $\alpha$th observation of a variate $z$ which takes on a finite number of integral values ranging from the negative integer $-r$ to the positive integer $m$ with respective probabilities $p_{-r}, \cdots, p_m$. Let $\xi_{ai} = P\lbrack Z_n = (a + i)\rbrack, (i = 1, 2, \cdots, m - 1)$, and $\xi_{bj} = P\lbrack Z_n = -(b + j)\rbrack, (j = 1, 2, \cdots, r - 1)$. Furthermore, let $A$ be a square matrix of $a + b - 1$ rows and columns with elements defined by: $a_{ii} = 1 - p_0$ for all $i; a_{i,i + k} = -p_k$ for $k = 1, 2, \cdots, m; a_{i,i- j} = -p_{-j}$ for $j = 1, 2, \cdots, r$; and $a_{ij} = 0$ otherwise. It is proved that \begin{equation*}\tag{i} \xi_{bj} = \sum^{r - j - 1}_{i=0} p_{i-r}A_{r-j-i,b},\quad(j = 0, 1, \cdots, r - 1) {(ii)} \xi_{aj} = \sum^{m - j - 1}_{i=0} p_{i+j +1}A_{a+b-i-1,b},\quad(j = 0, 1, \cdots, m - 1),\end{equation*} where $A_{kb}$ is the element of the $k$th row and $b$th column in $A^{-1}$. Let $E_{aj\tau}^n$ and $E_{bj\tau}^n$ be the conditional generating function of $n$ under the restriction that $Z_n = (a + j)$ and $Z_n = -(b + j)$ respectively. Then $\xi_{bj}E_{bj\tau}^n$ is obtained by substituting $\tau p_j$ for each $p_j$ occurring in equation (i) and $\xi_{aj}E_{aj\tau}^n$ is obtained by substituting $\tau p_j$ for each $p_j$ occurring in equation (ii). The probability that $Z_n = a + j$ in exactly $n$ steps is given by the coefficient of $\tau^n$ in the expansion of $\xi_{aj}E_{aj\tau}^n$ in a power series in $\tau$. The probability that $Z_n = -(b + j)$ in exactly $n$ steps is similarly obtained. This method is applied to the derivation of the exact power function and the distribution of $n$ for the sequential binomial probability ratio test. Part III. On Conjugate Distributions. Consider a random variable $X$ with a distribution density $f(x, \theta)$ which satisfies certain specified conditions. Let $\theta_1$ and $\theta_2$ be two values of $\theta$ and let $z = \log (f(x, \theta_2)/f(x, \theta_1))$. For any hypothesis $\theta = \theta'$, let $\varphi(t \mid \theta')$ be the moment generating function of $z$ and $h$ the non-zero value of $t$ for which $\varphi(t \mid \theta') = 1$. We set $F(x) = e^{hz}f(x, \theta')$. Then $f$ and $F$ are conjugate distributions. If $F = f(x, \theta")$, then $\theta'$ and $\theta"$ are defined as conjugate pairs. A method is given for obtaining the totality of conjugate pairs for the general class of distributions which admit a sufficient statistic. It is then shown that the power of the sequential probability ratio test based on such distributions is given explicitly in terms of these pairs. It is proven that within the approximation obtained by neglecting the excess of $\mid Z_n \mid$ over $a$ and $b$ at a decision point the following relationship holds: $P_b(n \mid F) = e^{-hb}P_b(n \mid f) \\ P_a(n \mid F) = e^{ha}P_a(n \mid f)$ where $P_b(n \mid g)$ and $P_a(n \mid g)$ stand for the probability that $Z_n \geq a$ and $Z_n \geq -b$ respectively in exactly $n$ steps under the hypothesis $g$.

The problem of estimating from a sample a confidence region for the parameters of the distribution of maximum values is treated by setting up what are called "statistical estimation functions" suggested by the functional form of the probability distribution of the sample, and finding the moment generating function of the probability distribution of these estimation "functions. Such an estimate by the method of maximum likelihood is also treated. A definition of "sufficiency" is proposed for "statistical estimation functions" analogous to that which applies to "statistics". Also the concept of "stable statistical estimation functions" is introduced. By means of a numerical illustration, four methods are discussed for setting up an approximate confidence interval for the estimated value of $x$ of the universe of maximum values which corresponds to a given cumulative frequency .99, for confidence level .95. Two procedures for solving this problem are recommended as practicable.

Consider a sequence $\{x_i\}$ of independent chance vectors in $k$ dimensions with identical distributions, and a sequence of mutually exclusive events $S_1, S_2, \cdots$, such that $S_i$ depends only on the first $i$ vectors and $\Sigma P(S_i) = 1$. Let $\varphi_i$ be a real or complex function of the first $i$ vectors in the sequence satisfying conditions: (1) $E(\varphi_i) = O$ and (2) $E(\varphi_j \mid X_1, \cdots, X_i) = \varphi_i$ for $j \geq i$. Let $\varphi = \varphi_i$ and $n = i$ when $S_i$ occurs. A general theorem is proved which gives the conditions $\varphi_i$ must satisfy such that $E\varphi = 0$. This theorem generalizes some of the important results obtained by Wald for $k = 1$. A method is also given for obtaining the distribution of $\varphi$ and $n$ in the problem of the "random walk" in $k$ dimensions for the case in which the components of the vector take on a finite number of integral values.

In using collateral information in an inverse probability situation to estimate a population fraction from a sample fraction it is necessary to use some particular form for the a priori probability function. This paper points out the advantages of using $Kx^r(1 - x)^s$ for this purpose. The application then involves only the Incomplete Beta Function. Graphs of the 10, 25, 50, 75 and 90 per cent points of the Incomplete Beta Function are given. They cover a range which includes and extends previous tabulations.

The enlargement principle provides techniques for inverting any nonsingular matrix by building the inverse upon the inverses of successively larger submatrices. The computing routines are relatively easily learned since they are repetitive. Three different enlargement routines are outlined: first-order, second-order, and geometric. None of the procedures requires much more work than is involved in squaring the matrix.

The joint frequency distribution has been found for any set of the $(n - k)$ deviates from their sample mean of each of the $t$ variates in a sample from a multivariate normal population. Expressions for the variance of any single deviate in this distribution, the correlation coefficient between any pair of deviates, and certain partial coerrelation coefficients between any pair have also been obtained. These results have been generalized so as to include the corresponding properties of deviates from a set of $t$ multiple linear regression equations estimated from the sample, the $m$ independent variates being the same for each of the $t$ dependent.

This note compares the power functions of the sign test for slippage with the power functions of the most powerful test for the case of normal populations. The sign test is found to be approximately 95% efficient for small samples.

It is shown that $\frac{1}{\sqrt{2\pi}} \int_{-x}^x e^{-\frac{1}{2}t^2} dt \geq \lbrack 1 - e^{-(2/\pi)x^2}\rbrack^\frac{1}{2}$ and that the equality is never in error by as much as three-fourths of one percent. Other approximations are discussed.