A new approach is presented to the problem of determining whether an individual (the target) appears in a large file where individuals are identified by measurements subject to error. This approach attaches costs to searching and to missing the individual. It corresponds to testing a simple hypothesis, that the measurements on the target and an element in the library have a given joint distribution, against the alternative that they are independent. Certain measures of information from large deviation theory are relevant. There is a surprising reduction in effectiveness of information in the presence of error. Data compression issues are studied. Attention is paid to a two-stage search procedure where the file is subdivided into piles which are in turn subdivided into bins. Each pile is examined and either discarded or searched. If it is searched, each bin in it is examined and either discarded or searched. If a bin is searched, each element of the bin is compared with the target.

This paper is a continuation of a paper in the Annals of Statistics (1976), 4 1159-1189 where, among other things, a Bayesian approach to testing independence in contingency tables was developed. Our first purpose now, after allowing for an improvement in the previous theory (which also has repercussions on earlier work on the multinomial), is to give extensive numerical results for two-dimensional tables, both sparse and nonsparse. We deal with the statistics $X^2, \Lambda$ (the likelihood-ratio statistic), a slight transformation $G$ of the Type II likelihood ratio, and the number of repeats within cells. The latter has approximately a Poisson distribution for sparse tables. Some of the "asymptotic" distributions are surprisingly good down to exceedingly small tail-area probabilities, as in the previous "mixed Dirichlet" approach to multinomial distributions (J. Roy. Statist. Soc. B. 1967, 29 399-431; J. Amer. Statist. Assoc. 1974, 69 711-720). The approach leads to a quantitative measure of the amount of evidence concerning independence provided by the marginal totals, and this amount is found to be small when neither the row totals nor the column totals are very "rough" and the two sets of totals are not both very flat. For Model 3 (all margins fixed), the relationship is examined between the Bayes factor against independence and its tail-area probability.

Sequential designs are proposed for clinical trials to compare two binomial success probabilities, $p_1$ and $p_2$. Approximations to the operating characteristics and expected sample size are obtained and compared with simulations. Special reference is made to the problem of comparing vasopressin and placebo for stopping upper gastrointestinal hemorrhage.

This paper considers the problem of sequentially estimating the mean of a normal distribution when the variance is unknown. A continuous time analogue of the discrete time problem is studied. For $L$ in a class of loss functions, properties of the value function and optimal continuation region of $L$ are presented. Asymptotic expansions are found for the value function and the optimal boundary function of the loss function $L$.

Suppose $X_1, X_2, \cdots$ are independent, identically distributed complex-valued $L^2$ random variables with $EX_1 = 0$ and $E(|X_1|^2) = 1$. Let $Y_{nk}$ be the $k$th Fourier coefficient of $X_1, \cdots, X_n$: $Y_{nk} = \sum^n_{j=1} X_j \exp \big(\frac{2\pi(-1)^{1/2}kj}{n}\big).$ Let $\mu_n$ be the empirical distribution of $\{n^{-1/2}Y_{nk}: k = 1, \cdots, n\}$. Then $\mu_n$ converges to the distribution of $U + iV$, where $U$ and $V$ are independent normal variables with mean 0 and variance $\frac{1}{2}$. This theorem is derived from a similar result for the Fourier coefficients of random permutations of the coordinates of $z^n$, where $z^n$ is a vector with $n$ coordinates such that $\max_k|z^n_k| = o(n^{1/2})$, as $n \rightarrow \infty$.

Robustness and efficiency of a parameter estimate $T$ can be assessed by comparing the fitted parametric distribution $P_T$ with the actual distribution, which is assumed to lie near the parametric family $\{P_\theta:\theta\in\Theta\}$. Asymptotic lower bounds are established for the minimax risk over distributions near the parametric model, taking as loss function a monotone increasing function of the Hellinger distance between the actual distribution of the sample and the fitted distribution determined by $T$. The set of marginal distributions considered in the minimax calculation is a subset of the Hellinger ball of radius $O(n^{-1/2})$ centered at $P_\theta, n$ being the sample size. When the loss function is bounded, the lower bound on maximum risk can be attained asymptotically. However, an estimator of $\theta$ which is asymptotically minimax for bounded loss functions may be far from optimal when the loss function is unbounded. Such divergent behavior is exhibited, for instance, by the sample mean in nearly normal models.

Invariance is used to show that Kariya and Eaton's test for multivariate spherical symmetry is UMP invariant against elliptically symmetric distributions. Also both the null and alternative distributions of the test statistic are found to be the same as those which occur when the sample is normally distributed. UMP and UMPU tests for serial correlation derived assuming normality are found to be even more robust against departure from this assumption than was recently demonstrated by Kariya. When applied to the linear regression model, these results give useful robustness properties for Kadiyala's $T1$ test and the Durbin-Watson test.

Two types of repeated measurements designs (RMD), the strongly balanced uniform RMD, and the balanced uniform RMD, are defined and their universal optimality is proved over very broad classes of competing designs. Construction methods for strongly balanced uniform RMD are also given.

We consider an experimental design problem in which $n$ treatments are applied successively to each experimental unit, and once applied their effects are permanent. To examine all $2^n - 1$ treatments combinations, a minimum of $\binom{n}{\big\lbrack \frac{n}{2} \big\rbrack}$ experimental units is both required and sufficient. A linear model is described and the first nontrivial case, $n = 4$, is examined in detail. It is shown that there are 24 nonisomorphic designs which reduce to 13 under the assumption of no interaction between the treatments. A serial correlation model is considered and the D, A and E, optimality criteria evaluated for $\rho = 0, 0.5$ and 0.75. Possible uses for the design automorphisms are then considered.

For the problem of weighing $k$ objects in $n$ weighings $(n \geq k)$ on a chemical balance, and certain related problems, we obtain new results and list the designs which have been proved $D$-optimum up to this time. While some of these optimality results have been known for some time, others are fairly recent. In particular, in the most difficult case $n \equiv 3(\operatorname{mod} 4)$ we prove a result characterizing optimum designs when $n \geq 2k - 5$. In addition, by a combination of theoretical bounds and computer search we find previously unknown optimum designs in the cases $(k, n) = (9, 11), (11, 15)$, and (12, 15), and establish the optimality of Mitchell's (10, 11) design. In some cases the optimum $X'X$ is not unique. Thus, we find two optimum $X'X$'s for the (6, 7), (8, 11), (10, 11), and (10, 15) cases. As a consequence of these results and other constructions, $D$-optimum designs are now known in all cases $k \leq 12$ (for all $n \geq k$), and in many other cases. Essentially complete listings for all $n \geq k$ had been given previously only for $k \leq 5$.

Let $S^d_k$ be the set of $d$th order splines on $\lbrack 0, 1 \rbrack$ having $k$ knots $\xi_1 < \xi_2 \cdots < \xi_k$. We consider the estimation of a sufficiently smooth response function $g$, using $n$ uncorrelated observations, by an element $s$ of $S^d_k$. For large $n$ and $k$ we have discussed the asymptotic behavior of the integrated mean square error (IMSE) for two types of estimators: (i) the least squares estimator and (ii) a bias minimizing estimator. The asymptotic expression for IMSE is minimized with respect to three variables. (i) the allocation of observation (ii) the displacement of knots $\xi_1 < \cdots < \xi_k$ and (iii) number of knots.

Let $Y:p \times r$ and $Z:p \times n$ be normally distributed random matrices whose $r + n$ columns are mutually independent with common covariance matrix, and $EZ = 0$. It is desired to test $\mu = 0$ vs. $\mu \neq 0$, where $\mu = EY$. Let $d_1, \cdots, d_p$ denote the characteristic roots of $YY'(YY' + ZZ')^{-1}$. It is shown that any test with monotone acceptance region in $d_1, \cdots, d_p$, i.e., a region of the form $\{g(d_1, \cdots, d_p)\leq c\}$ where $g$ is nondecreasing in each argument, is unbiased. Similar results hold for the problems of testing independence of two sets of variates, for the generalized MANOVA (growth curves) model, and for analogous problems involving the complex multivariate normal distribution. A partial monotonicity property of the power functions of such tests is also given.

This paper is concerned with jointly testing the hypotheses $\theta_i = \theta_{i0}, i = 1, \cdots, s$, using tests based on independent statistics $T_i$ with distributions $P(T_i \leq t) = F_i(t, \theta_i)$ nonincreasing in $\theta_i$. Holm proposed a sequentially rejective test procedure, applicable to this problem, for which, for fixed $\alpha (0 \leq \alpha \leq 1)$, the probability that the joint conclusion contains no false rejections is $\geq 1 - \alpha$ for all possible values of the $\theta_i$. Suppose, however, that if the hypothesis $\theta_i = \theta_{i0}$ is rejected, it is desired to conclude not only that $\theta_i \neq \theta_{i0}$ but also either that it is greater than $\theta_{i0}$ or smaller than $\theta_{i0}$. Usually one then requires a probability $\geq 1 - \alpha$ that the joint conclusion contains neither false rejections nor false directional statements. This paper considers the use of Holm's nondirectional procedure for rejecting hypotheses, supplemented by decisions on direction based on the values of the $T_i$. It is shown that this procedure does not in general provide the required control over error probabilities, but that it does so under specified conditions on the distributions of the $T_i$.

Let $d$ denote a positive integer, $\|x\| = (x^2_1 + \cdots + x^2_d)^{1/2}$ the Euclidean norm of $x = (x_1, \cdots, x_d) \in \mathbb{R}^d, k$ a nonnegative integer, $\mathscr{C}_k$ the collection of $k$ times continuously differentiable functions on $\mathbb{R}^d$, and $g_k$ the Taylor polynomial of degree $k$ about the origin corresponding to $g \in \mathscr{C}_k$. Let $M$ and $p > k$ denote positive constants and let $U$ be an open neighborhood of the origin of $\mathbb{R}^d$. Let $\mathscr{G}$ denote the collection of functions $g \in \mathscr{C}_k$ such that $|g(x) - g_k(x)| \leq M \|x\|^P$ for $x\in U$. Let $m \leq k$ be a nonnegative integer, let $\theta_0\in\mathscr{C}_m$ and set $\Theta = \{\theta_0 + g:g \in \mathscr{G}\}$. Let $L$ be a linear differential operator of order $m$ on $\mathscr{C}_m$ and set $T(\theta) = L\theta(0)$ for $\theta \in \Theta$. Let $(X, Y)$ be a pair of random variables such that $X$ is $\mathbb{R}^d$ valued and $Y$ is real valued. It is assumed that the distribution of $X$ is absolutely continuous and that its density is bounded away from zero and infinity on $U$. The conditional distribution of $Y$ given $X$ is assumed to be (say) normal, with a conditional variance which is bounded away from zero and infinity on $U$. The regression function of $Y$ on $X$ is assumed to belong to $\Theta$. It is shown that $r = (p - m)/(2p + d)$ is the optimal (uniform) rate of convergence for a sequence $\{\hat{T}_n\}$ of estimators of $T(\theta)$ such that $\hat{T}_n$ is based on a random sample of size $n$ from the distribution of $(X, Y)$. An analogous result is obtained for nonparametric estimators of a density function.

Edgeworth expansions for linear combinations of order statistics are established. The theorems require smooth weight functions and the underlying distribution function must possess a finite fourth moment. In addition a local smoothness condition is imposed on the underlying distribution function.

A very general result concerning the weak consistency and uniform asymptotic normality of the maximum likelihood estimator is presented. The result proves to be of particular value in establishing uniform asymptotic normality of randomly normalized maximum likelihood estimators of parameters in stochastic processes. The only conditions imposed are certain regularity conditions on the (random) information function, easily verified in practice. Application of the result is briefly considered.

It is customary to use a conditional test-statistic when testing for one parameter in a two-parameter exponential family model, the other parameter being considered as a nuisance parameter. In this note we discuss the Bahadur efficiency of such a conditional test and illustrate the results with examples from a branching process, a pure birth process, and a stable process.

In this paper it is shown that in the general case the exact noncentral distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ can be obtained in a very straightforward manner. This completely eliminates the need for the more complicated inverse Mellon transform. It is first shown that any random variable whose moments satisfy Wilks' Type B integral equation (Type B random variables) has a distribution that can be represented as a mixture of incomplete beta functions. Then it is shown that the moments of Wilks' $\Lambda$ and Wilks-Lawley $U$ criteria can be written as mixtures of the moments of Type B random variables. Combining these results yields the noncentral distribution of Wilks' $\Lambda$ and Wilks-Lawley $U$ criteria as mixtures of incomplete beta functions for the following tests: equality of two dispersion matrices; MANOVA; and canonical correlation.

Confidence bounds are derived for the upper 100$p$ percentile of an inverse distribution function. These bounds are considerably less conservative than similar bounds which apply to the entire function.

A Berry-Esseen theorem for $U$-statistics when the sample size is random is presented for the case when the random size is independent of the observations. This result extends the work of Callaert and Janssen. As an application of the special case of sample means, a rate of convergence to normality is obtained for the supercritical Galton-Watson process. Other possible applications are in sequential analysis.