The "goodness of fit" problem, consisting of comparing the empirical and hypothetical cumulative distribution functions (cdf's), is treated here for the case when an auxiliary parameter is to be estimated. This extends the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2]. The characteristic function of the limiting distribution of the test function is found by consideration of a Guassian stochastic process.

For the problem of testing a simple hypothesis on a density function of the form $f_\theta(e) = \exp \{\psi_0(\theta) + \sum^k_1 \psi_i(\theta)t_i(e) + t_0(e)\}$, explicit characterizations are given of a minimal essentially complete class of tests, the minimal complete class, and the closure of the class of Bayes' solutions, under certain assumptions. Applications are made to discrete distributions of the above form and to some problems of testing composite hypotheses. The likelihood ratio tests of these hypotheses are characterized and shown to be admissible under certain assumptions.

A compound decision problem consists of the simultaneous consideration of $n$ decision problems having identical formal structure. Decision functions are allowed to depend on the data from all $n$ components. The risk is taken to be the average of the resulting risks in the component problems. A heuristic argument for the existence of good asymptotic solutions was given by Robbins ([1] Sec. 6) and was preceded by an example (component decisions between $N(-1,1)$ and $N(1,1)$) exhibiting, for sufficiently large $n$, a decision function whose risk was uniformly close to the envelope risk function of "simple" decision functions. The present paper considers the class of problems where the components involve decision between any two completely specified distributions, with the risk taken to be the weighted probability of wrong decision. For all sufficiently large $n$, decision functions are found whose risks are uniformly close to the envelope risk function of "invariant" decision functions.

The efficiency of a family of tests is defined. Methods for evaluating the efficiency are discussed. The asymptotic efficiency is obtained for certain families of tests under assumptions which imply that the sample size is large.

A theorem by Pitman on the asymptotic relative efficiency of two tests is extended and some of its properties are discussed.

This paper describes sequential life test procedures, considering, as in a recent paper [4] devoted to nonsequential methods, the special case in which the underlying distribution of the length of life is given by the exponential density \begin{equation*}\tag{1} f(x, \theta) = e^{-x/\theta}/\theta,\quad x > 0.\end{equation*} The unknown parameter $\theta > 0$ can be thought of physically as the mean life. Our primary aim is to test the simple hypothesis $H_0: \theta = \theta_0$ against the simple alternative $H_1: \theta = \theta_1,$ where $\theta_1 < \theta_0,$ with type I and II errors equal to preassigned values $\alpha$ and $\beta,$ respectively. The test is carried out by drawing $n$ items at random from the population and placing them all on a life test. We consider both the replacement case, in which failed items are immediately replaced by new items, and the nonreplacement case. The test can be terminated either at failure times with rejection of $H_0$, or at any time between failures with acceptance of $H_0$. Since abnormally long intervals between failures furnish "information" in favor of $H_0$ and abnormally short intervals furnish "information" in favor of $H_1$, these features are not only reasonable but actually desirable. Similar problems involving a continuous time parameter have recently appeared [3], [5]. In this paper we obtain likelihood ratio tests and give approximate formulae for the O.C. (operating characteristic) curve, for the expected number of failures $E_\theta(r)$, and for the expected waiting time $E_\theta(t)$ before a decision is reached. In the replacement case where the number of items on test throughout the experiment is the same, namely $n$, it is shown that $E_\theta(t) = (\theta/n)E_\theta(r)$. A table giving approximate values of $E_\theta(r)$ for certain choices of $\theta_0/\theta_1, \alpha,$ and $\beta$ is given for the replacement case. Some calculations of exact $L(\theta)$ and $E_\theta(r)$ values using formulae in [1] and [3] are reported. Several numerical examples are worked out.

This paper suggests several `goodness of fit' test criteria, all having a linear form. The moment generating function and the limiting distribution of this linear form are obtained in Section 2. The best test criterion of this form for testing a simple hypothesis $H_0$ against a simple alternative hypothesis $H_1$ is shown, in Section 3, to be in general not independent of $H_1$. The remainder of this paper deals with a special case of the linear form, that is, the rank sum test criterion. The distribution of this test criterion is derived in Section 4, its consistency is proved in Section 5, and some numerical asymptotic efficiencies are calculated in Section 6. Within a certain class of tests, the present test is shown, in Section 7, to be uniformly most powerful for a special family of alternatives.

We consider here another aspect of a problem mentioned in a previous paper [2]. We shall be concerned with one-dimensional, real-valued random functions (r.f.) $X(t)$, defined for all $t$ and such that any sample taken at equidistant $t$-points satisfies a linear relation which is an analogue of one or other of the stochastic difference relations which are used for the analysis of discrete-parameter time-series. More specifically, we assume that there exist $k$ continuous and real-valued functions $\alpha_1(h), \cdots, \alpha_k(h) \text{of} h \geqq 0$ such that for any $h > 0$ and any $t$, the sequence \begin{equation*}\tag{1}\{X(t + \lbrack n + k \rbrack h) + \alpha_1(h)X(t + \lbrack n + k - 1 \rbrack h) + \cdots + \alpha_k(h)X(t + nh)\}, n = 0, \pm 1, \cdots,\end{equation*} satisfies certain conditions about independence or noncorrelation. In Section 1, we consider hypotheses concerning correlation, and find that the functions $\alpha_j(h)$ are restricted to certain forms. We also find that the assumption of zero serial correlations in the sequence (1) for all $h > 0$ implies that $X(t)$ is deterministic. In Section 2, we consider hypotheses concerning independence, and find the functions $\alpha_j(h)$ to be restricted as before.

Upper and lower bounds are obtained for the cumulative distribution function of the sample median of a sample of size $2n + 1$ drawn from a continuous population. It is shown that if the parent population is normal, then the distribution of the sample median tends "rapidly" to normality. Other kinds of parent populations are also discussed.

Three new test criteria are proposed for overall tests of hypotheses in multivariate analysis. They are based on the characteristic roots of certain matrices obtained from the product moment matrices of samples drawn from multivariate normal populations. The approximate distributions of the statistics involved in the tests are found as Type I or Type II Beta distributions.

A previous paper [1] has given a method of approximating the distribution of a quadratic form in normally distributed variables by means of convergent Laguerrian expansions. In the case of an indefinite quadratic form, however, the method was restrictive in that it might be difficult to obtain the semi-moments required in computing the coefficients of the expansion. The present article circumvents this difficulty for positive values of the argument of the distribution function, when the number of positive or the number of negative eigenvalues is even, and also yields convergent expansions for the distribution function involving Laguerre polynomials. The proposed method has the further advantage that no moments or semi-moments need be calculated.

An expression is derived for the distribution of a definite quadratic form in independent $N(0, 1)$ variates which depends only on the value of the determinant of the form and on the moments of a quadratic form whose matrix is the inverse of the original quadratic form. This expression is an alternating series which converges absolutely and is such that if we stop with any even power of the series we have an upper bound and if we stop with any odd power of the series a lower bound to the cumulative distribution function. The result given in this note seems to be in several ways an improvement over the method given in Robbins [2].

R. C. Bose and K. A. Bush [1] showed how to make use of the maximum number of points, no three of which are collinear, in finite projective spaces for the construction of orthogonal arrays. In particular this enabled them to construct an orthogonal array (81, 10, 3, 3). They proved on the other hand that, in the case considered, the maximum number of constraints does not exceed 12 [1], (Theorem 2C). Hence they state: "We do not know whether we can get 11 or 12 constraints in any other way." This paper shows first that a 10-rowed orthogonal array, constructed by the geometrical method, cannot be extended to an 11-rowed orthogonal array. It then shows that the number of constraints does not exceed 11. The problem of construction of an orthogonal array with 11 constraints remains unsolved. This summary should serve, as well, as a correction to the statement made in the abstract, "A remark on the geometrical method of construction of an orthogonal array," published in the Annals of Mathematical Statistics, Vol. 25 (1954), p. 177-178, which claimed the nonexistence of an orthogonal array of 11 constraints also.