This paper is concerned with optimum tests of certain composite hypotheses. In section 2 various aspects of a theorem of Scheffe concerning type $B_1$ tests are discussed. It is pointed out that the theorem can be extended to cover uniformly most powerful tests against a one-sided set of alternatives. It is also shown that the method for determining explicitly the optimum test region may in certain cases be reduced to a simple formal procedure. These results are used in section 3 to obtain optimum tests for the composite hypothesis specifying the value of the circular serial correlation coefficient in a normal distribution. A surprising feature of this example is the fact that for the simple hypothesis obtained by specifying values for the nuisance parameters no test with the corresponding optimum properties exists. In section 4 the totality of similar regions is obtained for a large class of probability laws which admit a sufficient statistic. Some composite hypotheses concerning exponential and rectangular distributions are treated in section 5. It is proved that the likelihood ratio tests of these hypotheses have various optimum properties.

This paper proposes a new test (the "quadrant sum") for the association of two continuous variables. Its notable properties are: (1) Special weight is given to extreme values of the variables. (2) Computation is very easy. (3) The test is non-parametric. Significance levels (for the quadrant sum) are given to the accuracy needed for practical use. To this accuracy they are independent of sample size (see Fig. 1). The generating function of the quadrant sum is given for the null hypothesis (no association = independence). A limiting distribution is deduced and compared with the cases $2n$ = 4, 6, 8, 10, and 14. Extension to higher dimensions and application to serial correlation are discussed.

Wald [2, 1943] extended the usefulness of tolerance limits to the simplest multi-dimensional cases. His principle is here used to provide many new ways of using a sample of $n$ to divide the range of the population into $n + 1$ blocks of known behavior. The exact tolerance distribution for the proportions of the population covered by these blocks is extended from the case of a continuous probability density function to the case of a continuous cumulative distribution function. Such an extension is needed in dealing completely with multivariate cases even where the underlying distribution is as smooth as a multivariate normal distribution. The devices used in Paper I [1] to extend the usefulness of tolerance limits to the case of a discontinuous underlying distribution will be applied in the next paper of this series, with some extension, to extend the usefulness of these general tolerance regions to the case of a discontinuous distribution. Some of these results specialize into new results for the univariate case, although they do not seem to have any immediate practical application. The author wishes to acknowledge the stimulation given to his work on this problem by Henry Scheffe, whose modesty has kept this paper from the joint authorship of papers I [1, Scheffe and Tukey 1945] and IV (not yet written).

In developing tests of fit based upon a sample $O_n(x_i)$ in the case that the cumulative distribution function $F(X)$ of the universe of $X$'s is not necessarily a function of a finite number of specific parameters--sometimes known as the non-parametric case--it has been pointed out by several writers that the "probability integral transformation" is a useful device (cf. [1]-[4]). The author finds that a modification of this approach is more effective. This modification is to use a transformation of ordered sample values $x_i$ from a random sample $O_n(x_i)$ based on successive differences of the cdf values $F(x_i)$. A theorem is proved giving a simple formula for the expected values of the products of powers of these differences, where all differences from 1 to $n + 1$ are involved in a symmetrical manner. The moment generating function of the test function defined as the sum of $m$ squares of these successive differences is developed and the application of such a test function is briefly discussed.

With any statistical decision procedure (function) there will be associated a risk function $r(\theta)$ where $r(\theta)$ denotes the risk due to possible wrong decisions when $\theta$ is the true parameter point. If an a priori probability distribution of $\theta$ is given, a decision procedure which minimizes the expected value of $r(\theta)$ is called the Bayes solution of the problem. The main result in this note may be stated as follows: Consider the class C of decision procedures consisting of all Bayes solutions corresponding to all possible a priori distributions of $\theta$. Under some weak conditions, for any decision procedure $T$ not in $C$ there exists a decision procedure $T^\ast$ in $C$ such that $r^\ast(\theta) \leqq r(\theta)$ identically in $\theta$. Here $r(\theta)$ is the risk function associated with $T$, and $r^\ast(\theta)$ is the risk function associated with $T^\ast$. Applications of this result to the problem of testing a hypothesis are made.

Given a set of $k$ random samples, $x_1, x_2, \cdots, x_k$, from a binomial distribution with parameters $p$ and $n$, it is shown that the familiar binomial index of dispersion $z = \frac{\sum^k_1 (x_i - \bar x)^2}{\bar x\big(1 - \frac{\bar x}{n_0}\big)}$ yields an approximate best critical region independent of $p$ for testing the hypothesis $n = n_0$ against the alternative hypothesis $n > n_0$, provided $\bar x$ and $n_0 - \bar x$ are not small. Because of the nature of the test, its optimum properties also apply to testing whether the data came from a binomial population with $n = n_0$ or from a Poisson population.

If a variable $x$ is normally distributed with mean zero, we have previously given a necessary and sufficient condition (see references at end of this paper) for the independence of two real symmetric quadratic forms in $n$ independent values of that variable. This condition is that the product of the matrices of the forms should vanish. In the present paper, we have proved that the same algebraic condition is both necessary and sufficient for the independence of two real symmetric bilinear, or a real symmetric bilinear and quadratic form, in normally correlated variables.

This paper discusses some of the concepts underlying small sample estimation and reexamines, in particular, the current notions on "unbiased" estimation. Alternatives to the usual unbiased property are examined with respect to invariance under simultaneous one-to-one transformation of parameter and estimate; one of these alternatives, closely related to the maximum likelihood method, seems to be new. The property of being unbiased in the likelihood sense is essentially equivalent to the statement that the estimate is a maximum likelihood estimate based on some distribution derived by integration from the original sampling distribution, by virtue of a "hereditary" property of maximum likelihood estimation. An exposition of maximum likelihood estimation is given in terms of optimum pairwise selection with equal weights, providing a type of rationale for small sample estimation by maximum likelihood.