The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument. These functions can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory, at least from an analytic point of view. From a computational and inference point of view, however, the hypergeometric functions are themselves of very limited value due primarily to the many difficulties involved in evaluating them numerically and consequently in studying the effects of population parameters on the distributions. Asymptotic results for large sample sizes or large population latent roots have so far proved to be much more useful for such problems. The purpose of this paper is to review some of the recent results obtained in these areas.

Causal effects are comparisons among values that would have been observed under all possible assignments of treatments to experimental units. In an experiment, one assignment of treatments is chosen and only the values under that assignment can be observed. Bayesian inference for causal effects follows from finding the predictive distribution of the values under the other assignments of treatments. This perspective makes clear the role of mechanisms that sample experimental units, assign treatments and record data. Unless these mechanisms are ignorable (known probabilistic functions of recorded values), the Bayesian must model them in the data analysis and, consequently, confront inferences for causal effects that are sensitive to the specification of the prior distribution of the data. Moreover, not all ignorable mechanisms can yield data from which inferences for causal effects are insensitive to prior specifications. Classical randomized designs stand out as especially appealing assignment mechanisms designed to make inference for causal effects straightforward by limiting the sensitivity of a valid Bayesian analysis.

The "procedures" discussed in this paper are of the following type: The statistician makes a conventional decision (in a multiple decision problem). He also provides a statement of the guaranteed conditional probability that his decision will be correct, given the observed value of some conditioning random variable. Various admissibility criteria to relate such procedures are proposed. For example, we say that one procedure is better (first sense) than a second if the guaranteed conditional confidence statement using the first is stochastically larger than that using the second, for all possible states of nature. Some ramifications of these admissibility criteria are discussed, and some specific admissible procedures are described for problems with two possible states of nature. In particular, the procedure having the finest possible monotone conditioning and having equal conditional confidence under both states of nature is shown to have many desirable admissibility properties.

We study the tail of the null distribution of the $\log$ likelihood ratio statistic for testing sharp hypotheses about the parameters of an exponential family. We show that the classical chisquare approximation is of exactly the right order of magnitude, although it may be off by a constant factor. We then apply our results and techniques to find the error probabilities of a sequential version of the likelihood ratio test. The sequential version rejects if the likelihood ratio statistic crosses a given barrier by a given time. Our approach uses a local limit theorem which takes account of large deviations and integrates the local result by using the theory of Hausdorff measures.

A general theorem which is useful in proving the exponential boundedness of the stopping time of sequential tests for parameters in general linear models is formulated; this theorem is formulated under the assumptions that the squared error has a finite moment-generating function and the sequence of the running averages of the concomitant variables converges. Applications are given.

In comparing two treatments, eligible subjects come to the experiment sequentially and must be treated at once. To reduce experimental bias and to increase the precision of inference about treatment effects, the adaptive biased coin design, which offers a compromise between perfect balance and complete randomization, is proposed and analyzed. This new design has the property that it forces a small-sized experiment to be balanced, but tends toward the complete randomization scheme as the size of the experiment increases.

Statistical properties are derived for maximum likelihood estimates of dose-response functions in which the response probability is related to the dose by means of a polynomial of unknown degree with nonnegative coefficients. Dose-response functions of this form are predicted by the multistage model of carcinogenesis. We first establish necessary and sufficient conditions for strong consistency of the estimates. For these results no assumptions are made about the polynomial degree, so the number of coefficients to be estimated is effectively infinite. Under some additional assumptions, which do involve restrictions on the polynomial degree, we obtain the asymptotic distribution of the vector of maximum likelihood estimates about the true vector of polynomial coefficients. Because the coefficients are constrained to be nonnegative, the limiting distribution will generally not be normal.

In the analysis of life tables one biometric function of interest is the life expectancy at age $x, e_x = E\lbrack X - x\mid X > x\rbrack$. Estimation of $e_x$ is considered, the standard estimator used in life tables is shown to be asymptotically unbiased, uniformly strong consistent, and converges in distribution to a Gaussian process. The connections of the estimator studied in this article and that used in reliability theory are illustrated.

A set of observations is partitioned into $k$ clusters by optimizing a clustering criterion $W$. The asymptotic distribution of this clustering criterion may be determined simply in certain cases where the optimal sample partition differs negligibly from the optimal population partition. Detailed proofs are given in the one-dimensional case when the clustering criterion to be minimized is within cluster sum of squares. The asymptotic distributions are used to compute approximate significance levels of tests for the presence of clusters, and of tests for bimodality.

Let $\mathscr{F}_1, \mathscr{F}_2$ be two families of $p$-variate distribution functions with specified means $\mathbf{\mu}_i (i = 1,2)$ and nonsingular covariance matrices $\Sigma_i$, and let $\pi_i$ be the prior probability assigned to $\mathscr{F}_i$ for $i = 1, 2$. The objective is to discriminate whether an observation $\mathbf{x}$ is from a distribution $F_1 \in \mathscr{F}_1$ or $F_2 \in \mathscr{F}_2$. Given a pair $F = (F_1, F_2)$ the error probability for classification rule $\phi$ is denoted by $e(\phi, F)$. In this paper the values of $\sup_F \inf_\phi e(\phi, F)$ and $\inf_\phi \sup_F e(\phi, F)$ are found and conditions for the existence of a saddle point of $e(\phi, F)$ are given. Also a saddle point is found when it exists. When $\phi$ is restricted to linear classification rules the same problems are considered. The mathematical programming method for finding a saddle point is also outlined.

Suppose that a sample of size $n$ from a distribution function $F$ is obtained. However, only $r(< n)$ values from the sample are observed, say $X_1,\cdots, X_r$. Without loss of generality, we can consider $X_1,\cdots, X_r$ to be the first $r$ values in the (unordered) sample. The problem is to estimate the rank order $G$ of $X_1$ among $X_1,\cdots, X_n$. The situations of interest include $F$ nonrandom, either known or unknown, and $F$ random. The random case assumes that $F$ is a random distribution function chosen according to Ferguson's (Ann. Statist. 1 (1973) 209-230) Dirichlet process prior. Since this random distribution function is discrete with probability one, average ranks are used to resolve ties. A Bayes estimator (squared-error loss) of $G$ is developed for the random model. For the nonrandom distribution function model, optimal non-Bayesian estimators are developed in both the case where $F$ is known and the case where $F$ is unknown. These estimators are compared with the Dirichlet estimator on the basis of average mean square errors under both the random and nonrandom models.

For a simple regression model, the problem of estimating the intercept after a preliminary test on the regression coefficient is considered here. Some nonparametric procedures for this problem are formulated and their various asymptotic properties studied. Comparison with the conventional estimation procedures (for both the situations where the regression coefficient is treated as a nuisance parameter or not) has also been made.

The very general results of Beran and Gine on invariant tests of uniformity are applied to $S_p$, the surface of the unit hypersphere, and $H_p$, the surface of the unit hypersphere with antipodes identified, to give a class of invariant tests of uniformity for signed and unsigned directional data in $(p + 1)$-dimensions. The $(p + 1)$-dimensional analogues of the test statistics due to Rayleigh, Bingham, Ajne, and Gine are constructed as the simplest examples, and corresponding methods are derived for particular orientation statistics as examples on $H_3$.

The estimation of a density and its derivatives by the kernel method is considered. Uniform consistency properties over the whole real line are studied. For suitable kernels and uniformly continuous densities it is shown that the conditions $h \rightarrow 0$ and $(nh)^{-1} \log n \rightarrow 0$ are sufficient for strong uniform consistency of the density estimate, where $n$ is the sample size and $h$ is the "window width." Under certain conditions on the kernel, conditions are found on the density and on the behavior of the window width which are necessary and sufficient for weak and strong uniform consistency of the estimate of the density derivatives. Theorems on the rate of strong and weak consistency are also proved.

The limit distribution of the least squares estimator $\hat{\alpha}$ of the parameter $\alpha$ of the first order stochastic difference equation, in the boundary case $|\alpha| = 1$, is presented. With this, the asymptotic distributional problem for any real $\alpha$ in the first order case is completely settled.

For an infinite sequence of independent and identically distributed random variables $X, X_1, X_2,\cdots; X_n,\cdots$ for which $EX = 0$ and $\operatorname{Var} X = 1$, the behavior of crossings that $S_n = \sum_{1\leqq k\leqq n} X_k \geqq n\varepsilon (\varepsilon > 0)$ for some $n \geqq 1$ has recently been under intensive investigation, of which the subject matters are the largest excess $Z = \sup_{n\geqq 1} (S_n - n\varepsilon)^+$, the last time $M = \sup \{n; S_n \geqq n\varepsilon, 1 \leqq n < \infty\}$, or $= 0$ if no such $\sup$ exists, and the number of crossings $N = \sum_{1\leqq n<\infty} I\{S_n \geqq n\varepsilon\}$ ($I$ means the indicator function). This paper describes a striking distributional similarity between $\varepsilon^2N$ and $\varepsilon^2(M - N)$ in the limiting sense as $\varepsilon \rightarrow 0$. Moreover, a new and systematic treatment for the moments problem unifies the previously published results as well as giving some new results. Existence of the limiting moments as $\varepsilon \rightarrow 0$ and of the moment generating function is also considered in detail. Most of the results for the one-sided crossings (i.e., $S_n \geqq n\varepsilon$) are then extended to cover their analogues in two-sided crossings (i.e., $|S_n| \geqq n\varepsilon)$.

This paper treats a generalization of the classical MANOVA testing problem. The problem is reduced via invariance considerations and a new test statistic is proposed. This new test is shown to be a unique locally best invariant test and locally minimax.

A number of statistics that arise in time series analysis can be represented as the sum of a partial realization of a possibly serially dependent and nonstationary discrete-parameter stochastic process. The almost sure and $L_p, p > 1$, convergence of such statistics is investigated, under various moment conditions. The results are applied to the least squares estimates of multiple regressions.

Let $\{x_t\}$ be a purely nondeterministic stationary process satisfying all the assumptions made by Berk (1974), and $\{y_t\}$ be another purely nondeterministic stationary process. Assume that $y_t$ is independent of $x_t$ but has exactly the same statistical properties as that of $x_t$. Consider the linear prediction of future values of $y_t$ on the basis of past values, using prediction constants estimated from a realisation of $T$ observations of $x_t$ by least-squares fitting of an autoregression of order $k$. By assuming that $k \rightarrow \infty, k^3/T \rightarrow 0$ as $T \rightarrow \infty$, the effect on the mean square error of prediction of estimating the autoregressive coefficients is determined. This effect is the same as for the case when the prediction constants are estimated by factorising a "windowed" estimate of the spectral density function of $x_t$.

A counterexample is given to a proposition of Marshall and Olkin (1974, Ann. Statist.) describing a probability bound. A theorem which gives the conditions under which mixtures of associated random variables remain associated, is stated. This provides a method to obtain the required bound.

It is shown that the characteristic roots of a Wishart matrix (identity covariance matrix) and the roots of $S_1 S_2^{-1}$ and $S_1(S_1 + S_2)^{-1}$ where $S_1, S_2$ are independent $p \times p$ Wishart matrices with the same covariance matrix, satisfy certain types of dependency relationships. That is, it is shown that these roots are (a) positive orthant dependent, (b) associated, (c) stochastically increasing in sequence, and (d) positively likelihood ratio dependent. An example of how this may be used in obtaining simultaneous confidence intervals is also included.