A question of interest in connection with many statistical problems is the following: Does a slight change in the problem result in a different answer? Here the effect of changes in the testing problem on the minimal complete class of tests is investigated. The effects of such changes are found to be different for the two families of distributions considered: The discrete multivariate exponential family and the continuous multivariate exponential family. In Section 2, it is shown that with respect to the discrete exponential family, the minimal complete class of tests for a standard testing problem is minimal complete for a wide variety of related problems. In Section 3, an example is given showing that with respect to the continuous exponential family, on the other hand, the minimal complete class of tests for a standard problem is not necessarily minimal complete for a slight variation of this problem. Tests that are admissible for the standard problem are not necessarily admissible for the variation. Partly in a general decision theoretic framework and partly with respect to specific examples, Hoeffding [2] has discussed the effect of changes in the family of probability distributions on the minimax solution and other optimal solutions. He has also given key references to the extensive literature on the performance of standard procedures for families of probability distributions not satisfying all the assumptions under which the standard procedures were derived. Workers in this area have primarily concentrated on the effect of changes in the probability model on a single solution rather than on a class of solutions, for example, the class of admissible solutions, as we do here. We recall some basic ideas. Consider the probability structure $(\mathcal{X, A,} P, \Omega)$ where $\mathcal{X}$ and $\Omega$ are sets, $\mathcal{A}$ is a $\sigma$-field of subsets of $\mathcal{X}$, and for each $\theta$ in $\Omega$, $P_{\theta}$ is a probability measure on $\mathcal{A}$. Relative to the above structure, a testing problem is an ordered pair $(\omega_0, \omega_1)$ of disjoint subsets of $\Omega$. A test $\varphi$ is a function from $\mathcal{X}$ into $\lbrack 0, 1\rbrack$ measurable with respect to $\mathcal{A}$. The test $\varphi$ is used in the following way: A random element $X$ with values in $\mathcal{X}$ having $P_{\theta}$ as its probability distribution is observed. If $x$ is the outcome then the hypothesis $H: \theta \varepsilon \omega_0$ is rejected with probability $\varphi(x)$ in favor of the alternative $A: \theta \varepsilon \omega_1$. If the test $\varphi$ is used and $\theta$ is the parameter, then the probability that $H$ is rejected is $E_{\theta\varphi} = \int_{\mathcal{X}} \varphi(x) dP_{\theta}(x)$. If $\varphi$ and $\varphi^{\ast}$ are tests, then $\varphi$ is at least as good as $\varphi^{\ast}$ if $E_{\theta\varphi} - E_{\theta\varphi^\ast} \leqq 0$ for $\theta$ in $\omega_0, \geqq 0$ for $\theta$ in $\omega_1$. The test $\varphi$ is better than $\varphi^{\ast}$ if $\varphi$ is at least as good as $\varphi^{\ast}$ but $\varphi^{\ast}$ is not at least as good as $\varphi$. A test $\varphi$ is admissible if there is no test better than $\varphi$. A class of tests is complete if to each test not in the class there corresponds a test in the class which is better. A class is minimal complete if it is complete and no proper subclass is complete. The notions, "essentially complete" and "minimal essentially complete," are defined similarly with "at least as good" substituted for "better."

Let $X$ be a normal random variable with variance one and mean either $\pm a/2$, where $a$ is a given positive constant. Let $C_m(m \geqq 0)$ denote the class of all two-stage rules with first sample size of $m$ for deciding, after two successive samples of independent observations on $X$, which of the two mean values is correct. This paper investigates the class of Bayes rules in $C_m$, parametrized by a priori probabilities on the hypotheses, and simple wrong decision losses. The cost per observation is taken throughout to be unity. Section 1 gives some general properties of Bayes rules in $C_m$ for decisions between any two continuous densities for $X$; Sections 2, 3, and 4 concern the densities specified above. Section 2 consists of a detailed development of Bayes second sample size properties in terms of the Bayes parameters and first sample outcomes. For example, Theorem 2.1 gives non-trivial lower and upper bounds for positive values of the Bayes second sample size corresponding to any fixed value for the minimum wrong decision loss. In Section 3, sufficient conditions are given under which the losses may be chosen so as to obtain Bayes rules with preassigned invariant error probabilities. (Invariance is taken with respect to changes in the prior probabilities.) It is shown how this result leads to rules which minimize the maximum expected sample size among rules in $C_m$ with error probabilities less than or equal to specified values. An illustrative example is considered for the case when these specified bounds are equal. The selection of an optimum first sample size for this example is treated in Section 4. The resulting rule has the above described good property among all two-stage rules (of any first sample size) subject to these bounds. Tables are included giving optimum first and second sample sizes and the values of auxiliary functions when this common bound on the error probabilities is .05 and .01.

Sections 1-6 are concerned with lower bounds for the expected sample size, $E_0(N)$, of an arbitrary sequential test whose error probabilities at two parameter points, $\theta_1$ and $\theta_2$, do not exceed given numbers, $\alpha_1$ and $\alpha_2$, where $E_0(N)$ is evaluated at a third parameter point, $\theta_0$. The bounds in (1.3) and (1.4) are shown to be attainable or nearly attainable in certain cases where $\theta_0$ lies between $\theta_1$ and $\theta_2$. In Section 7 lower bounds for the average risk of a general sequential procedure are obtained. In Section 8 these bounds are used to derive further lower bounds for $E_0(N)$ which in general are better than (1.3).

Estimators of mean and standard deviation for censored normal samples which are based on linear systematic statistics and which use simple coefficients are almost as efficient as estimators using the best possible coefficients. Estimators are given for samples of size $N \leqq 20$ for censoring at one extreme and for several types of censoring at both extremes.

In sampling from a finite population, the nonexistence of a uniformly minimum variance unbiased estimator for the mean $\mu$ has been demonstrated by Godambe [3], and the inadmissibility of the sample mean as an estimator for $\mu$, when sampling is with replacement and equal probabilities, has been proved by Des Raj and Khamis [2] and by Basu [1]. In this paper, the problem of unbiased linear estimation of $\mu$ with minimum variance is considered for a very general scheme of sampling. An admissible estimator is obtained, together with a complete class of estimators. It is shown further that, for a somewhat restricted sampling scheme, amongst estimators with variance proportional to $\sigma^2$, there does exist a best estimator which, in the case of sampling with replacement and equal probabilities, is the same as that considered in [1] and [2].

It has been suggested ([5], [6]) to use the expected value, $E(l)$, of the length, $l$, of a confidence interval for the variable to be predicted as a measure of precision of prediction in a regression analysis. The measure is relevant only if the predictor variables are random. Criticism of this particular choice of a measure of precision usually centres about the following questions: a. Why is the measure based on this and no other system of confidence intervals; what is known about optimality of this system. (The system referred to will be described in Section 2.) b. Why is it base don the physical length of the intervals and not on Neyman's "shortness"? c. If it is agreed that it is based on $l$, what justifies the choice of $E(l)$? In the following a few points are raised which are, of course, not sufficient to prove that $E(l)$ is the only possible choice for a measure of precision, but which indicate that the intuitive choice is not altogether unreasonable. Not much can be said about a. It turns out, in Section 3, that the confidence limits discussed here are unbiased, but nothing about optimality in any sense is known to the author. To throw some light on b, Neyman's shortness of the system of confidence intervals used is calculated in Section 3. A parameter enters the problem which makes it impossible to use Neyman's shortness as an overall measure of precision. With regard to c, one may argue that $l$ is a random variable and if a single measure of precision is needed a single characteristic of its distribution must be used. Obvious possibilities are the mean and the median. The distribution of $l$ is obtained in Section 4, and it becomes apparent that the use of the median would involve heavy numerical calculations.

Asymptotic expressions up to terms of order $n^{-2}$ are given for the efficiency of the Wilcoxon two-sample test relative to the standard-normal test and $t$-test for nearby alternatives. The first term is the well-known Pitman efficiency; the remaining terms are corrections for finite sample sizes. Efficiency values are given for finite sample sizes in the case of normal and rectangular distributions, and comparisons of the asymptotic with the exact efficiency values are made. In general, the Wilcoxon test is shown to be nearly as good locally for moderate sample sizes as it is known to be asymptotically. A similar analysis is performed for the single-sample sign test.

Rank order theory is developed for the two-sample problem in which censoring of the observations has occurred, i.e., not all of the random variables are observed. The approach is similar to [2] with the striking difference that in the present case the rank orders are not all equally likely under the null hypothesis, and thus it becomes important to work with the likelihood ratios of rank orders. In applying the results of this paper, there will be a strong analogy to sequential analysis. The censoring scheme corresponds to the stopping rule and in both cases the terminal decision should be based on the likelihood ratio. We do not give the detailed applications of the present theory either to earlier procedures or to the new ones introduced here.

Rosenblatt's derivation [3] of the limiting distribution of the statistic (1) below contains an incorrect step. A simple argument is presented that corrects Rosenblatt's proof, so that his conclusion is shown to be valid.

Properties of the sequence of random variables $-(1/n) \log p$ are obtained for an arbitrary, not necessarily ergodic or stationary, information source. These permit an elementary combinatorial proof of the AEP (asymptotic equipartition property).

This paper is concerned with the waiting time process, $W(t)$, for the queueing system in which (1) there is only one counter, (2) the customers arrive at random and are served in the order of arrival, and (3) the service time distribution has a general form. It is observed that the Pollaczek-Khintchine formula for the transform of the limiting distribution of $W(t)$ is similar to the one occurring in the theory of continuous time storage processes, and it is inverted by the method used in that theory. Further, $W(t)$ is shown to be a special case of the storage process, and known methods and results of the storage theory are used to obtain the transition distribution function of $W(t)$.

It is well known (see e.g. [4] p. 198) that for every positive $\epsilon$ and every square integrable random variable $X$ with zero expectation, $P{X \geqq \epsilon} \leqq E (X^2)/\lbrack\epsilon^2 + E(X^2)\rbrack$. In this paper an inequality is obtained that generalizes this in the same way that Kolmogorov's inequality generalizes Chebyshev's inequality. The inequality is proved in Section 2 and an example is given to show that equality can be achieved. In Section 3 an extension to continuous parameter martingales is obtained, and a condition under which equality can be achieved is given.

If $x$ is a random variable with mean zero and variance $\sigma^2$, then, according to Chebyshev's inequality, $P{|x| \geqq 1} \leqq \sigma^2$. The corresponding one-sided inequality $P{x \geqq 1} \leqq \sigma^2/(\sigma^2 + 1)$ is also known (see e.g. [2, p. 198]). Both inequalities are sharp. A generalization of Chebyshev's inequality was obtained by Olkin and Pratt [1] for $P{|x_1| \geqq 1 or \cdots or |x_k| \geqq 1}$, where $Ex_i = 0, Ex_i^2 = \sigma^2$, $Ex_ix_j = \sigma^2 \rho (i \neq j), i, j = 1, \cdots, k;$ we give here the corresponding generalization of the one-sided inequality, and we consider also the case where only means and variances are known. To obtain an upper bound for $P\{x \varepsilon T\} \equiv P\{x_1 \geqq 1 \text{or} \cdots \text{or} x_k \geqq 1\}$, we consider a nonnegative function, $f(x) \equiv f(x_1, \cdots, x_k)$, such that $f(x) \geqq 1$ for $x \varepsilon T$. Then $Ef(x) \geqq \int_{\{x \varepsilon T\}} f(x) dP \geqq P\{x \varepsilon T\}$. Since the bound is to be a function of the covariance matrix, $\Sigma, f(x)$ must be of the form $(x - a)A(x - a)'$, where $a = (a_1, \cdots, a_k), A = (a_{ij}): k \times k$. A "best" bound is one which minimizes $Ef(x) = \operatorname{tr} A(\Sigma + a'a)$, subject to $f(x) \geqq 0, f(x) \geqq 1$ on $T$.

Connor [3] has shown that the relations among the parameters of the triangular association scheme themselves imply the scheme if $n \geqq 9$. This result was shown by Shrikhande [6] to hold also if $n \leqq 6$. (The problem has no meaning for $n < 4$.) This paper shows that the result holds if $n = 7$, but that it is false if $n = 8$.

The proper spaces of the matrix $NN'$, where $N$ is the incidence matrix of a triangular partially balanced incomplete block design, are exhibited explicitly; this provides a convenient form for the Gramian of a basis of the join of two of these spaces.

Usually, in a factorial experiment, the block size of the experiments is not large enough to permit all possible treatment combinations to be included in a block. Hence we resort to the theory of confounding. With respect to symmetric factorial designs, the theory of confounding has been highly developed by Bose [1], Bose and Kishen [2] and Fisher [4], [5]. An excellent summary of the results of this research appears in Kempthorne [6]. Some examples of asymmetric factorial designs can be found in Yates [14], Cochran and Cox [3], Li [8], Kempthorne [6] and Nair and Rao [9], [10]. Nair and Rao [11] have given the statistical analysis of a class of asymmetrical two-factor designs in considerable detail. The author [13] has considered the problem of achieving "complete balance" over various interactions in factorial experiments. In the present paper a class of factorial experiments, balanced factorial experiments (BFE) (Definition 4.2) is considered. The theorems proved in Section 4 outline a detailed analysis of BFE's, including estimates of various interactions at different levels. Finally, a method of constructing BFE's is given in Section 6. It should be noted that Theorems 5.2 to 5.5 are generalisations of the corresponding theorems by Zelen [15], and the method of construction in Section 6 is a general form of the one indicated by Yates [14], Nair and Rao [9], [10] and Kempthorne [6] (Section 18.7).

The inequality $b \geqq v + r - 1$ for a balanced incomplete block design was proved by Bose [1] under the assumption of resolvability. In this note the inequality is proved without that assumption, but with the weaker assumption that $v = nk$.