The Lugannani and Rice formula for tail areas in the univariate case has recently been extended to tail areas for marginal distributions and for conditional distributions in certain multivariate settings. However, the results on relative order of errors are given only formally or under the strong continuity assumptions necessary to obtain density approximations. This paper attempts to give a unified treatment of these results for smooth transformations of multivariate means under weaker conditions appropriate to indirect Edgeworth approximations for probabilities.

Motivated by the first-order Pitman closeness of best asymptotically normal estimators and some recent developments on higher-order asymptotic efficiency of estimators, a second-order asymptotic theory is developed for comparison of estimators under the Pitman closeness criterion, covering both the cases without and with nuisance parameters. The notion of second-order Pitman admissibility is also developed.

Extending work of Rubin, this paper explores a Bayesian counterpart of the classical $p$-value, namely, a tail-area probability of a "test statistic" under a null hypothesis. The Bayesian formulation, using posterior predictive replications of the data, allows a "test statistic" to depend on both data and unknown (nuisance) parameters and thus permits a direct measure of the discrepancy between sample and population quantities. The tail-area probability for a "test statistic" is then found under the joint posterior distribution of replicate data and the (nuisance) parameters, both conditional on the null hypothesis. This posterior predictive $p$-value can also be viewed as the posterior mean of a classical $p$-value, averaging over the posterior distribution of (nuisance) parameters under the null hypothesis, and thus it provides one general method for dealing with nuisance parameters. Two classical examples, including the Behrens-Fisher problem, are used to illustrate the posterior predictive $p$-value and some of its interesting properties, which also reveal a new (Bayesian) interpretation for some classical $p$-values. An application to multiple-imputation inference is also presented. A frequency evaluation shows that, in general, if the replication is defined by new (nuisance) parameters and new data, then the Type I frequentist error of an $\alpha$-level posterior predictive test is often close to but less than $\alpha$ and will never exceed $2\alpha$.

The definition and elementary properties of Polya tree distributions are reviewed. Two theorems are presented showing that Polya trees can be constructed to concentrate arbitrarily closely about any desired pdf, and that Polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy, thereby satisfying a consistency condition. Such theorems are false for Dirichlet processes. Models are constructed combining partially specified Polya trees with other information such as monotonicity or unimodality. It is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications. A numerical example is given. A theorem of Diaconis and Freedman about Dirichlet processes is generalized to Polya trees, allowing Polya trees to be the models for errors in regression problems. Finally, empirical Bayes models using Dirichlet processes are generalized to Polya trees. An example from Berry and Christensen is reanalyzed with a Polya tree model.

Formal posteriors for improper priors are investigated in connection with coherence, both in the sense of Regazzini and of Heath and Sudderth. Those priors $\pi$ which are linked with the improper prior $\gamma$ by the relation $\pi(B) = 0$ whenever $\gamma(B) < +\infty$ are studied in particular. Moreover, a characterization of the inferences which are coherent according to Heath and Sudderth is found, and several examples, exhibiting several phenomena, are given.

Estimation of a covariance matrix $\sum$ is a notoriously difficult problem; the standard unbiased estimator can be substantially suboptimal. We approach the problem from a noninformative prior Bayesian perspective, developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting Bayes estimators. These expressions involve the computation of high-dimensional posterior expectations, which is done using a recent Markov chain simulation tool, the hit-and-run sampler. Frequentist risk comparisons with previously suggested estimators are also given, and determination of the accuracy of the estimators is addressed.

Sequential selections are to be made from two independent stochastic processes, or "arms." At each stage we choose which arm to observe based on past selections and observations. The observations on arm $i$ are conditionally i.i.d. given their marginal distribution $P_i$ which has a Dirichlet process prior with parameter $\alpha_i, i = 1, 2$. Future observations are discounted: at stage $m$, the payoff is $a_m$ times the observation $Z_m$ at that stage. The discount sequence $A_n = (a_1, a_2,\cdots, a_n, 0,0,\cdots)$ is a nonincreasing sequence of nonnegative numbers, where the "horizon" $n$ is finite. The objective is to maximize the total expected payoff $E(\sum^n_1a_iZ_i)$. It is shown that optimal strategies continue with an arm when it yields a sufficiently large observation, one larger than a "break-even observation." This generalizes results of Clayton and Berry, who considered two arms with one arm known and assumed $a_m = 1 \forall m \leq n$.

A general missing information principle is proposed for constructing $M$-estimators of regression parameters in the presence of left truncation and right censoring on the observed responses. By making use of martingale central limit theorems and empirical process theory, the asymptotic normality of $M$-estimators is established under certain assumptions. Asymptotically efficient $M$-estimators are also developed by using data-dependent score functions. In addition, robustness properties of the estimators are discussed and formulas for their influence functions are derived for the robustness analysis.

We study the nonparametric maximum likelihood estimator (NPMLE) for a concave distribution function $F$ and its decreasing density $f$ based on right-censored data. Without the concavity constraint, the NPMLE of $F$ is the product-limit estimator proposed by Kaplan and Meier. If there is no censoring, the NPMLE of $f$, derived by Grenander, is the left derivative of the least concave majorant of the empirical distribution function, and its local and global behavior was investigated, respectively, by Prakasa Rao and Groeneboom. In this paper, we present a necessary and sufficient condition, a self-consistency equation and an analytic solution for the NPMLE, and we extend Prakasa Rao's result to the censored model.

Proportional hazards models with stochastic baseline hazards and estimators of the relative risk coefficient in these models were proposed by Prentice, Williams and Peterson and by Chang and Hsiung in medical and industrial contexts. The form of the estimating functions recommended varies according to the form of the unknown stochastic baseline hazards. This paper examines the same estimation problem in the context of large-sample theory. It is shown that the proposed estimators are regular, asymptotically normal and asymptotically efficient. Asymptotic information and representation theorems in the sense of Begun, Hall, Huang and Wellner are also provided for these models.

We consider the problem of estimating the bivariate distribution of the random vector $(X, Y)$ when $Y$ may be subject to random censoring. The censoring variable $C$ is allowed to depend on $X$ but it is assumed that $Y$ and $C$ are conditionally independent given $X = x$. The estimate of the bivariate distribution is obtained by averaging estimates of the conditional distribution of $Y$ given $X = x$ over a range of values of $x$. The weak convergence of the centered estimator multiplied by $n^{1/2}$ is obtained, and a closed-form expression for the covariance function of the limiting process is given. It is shown that the proposed estimator is optimal in the Beran sense. This is similar to an optimality property the product-limit estimator enjoys. Using the proposed estimator of the bivariate distribution, an extension of the least squares estimator to censored data polynomial regression is obtained and its asymptotic normality established.

Suppose we observe $Y-i = f(x_i) + \varepsilon_i, i = 1, \ldots, n$. We wish to find approximate $1 - \alpha$ simultaneous confidence regions for $\{f(x), x \in \mathscr{X}\}$. Our regions will be centered around linear estimates $\hat{f}(x)$ of nonparametric or nonparametric $f(x)$. There is a large amount of previous work on this subject. Substantial restrictions have been usually placed on some or all of the dimensionality of $x,$ the class of functions $f$ that can be considered, the class of linear estimates $\hat{f}$ and the region $\mathscr{X}$. The method we present is an approximation to the tube formula dn can be used for multidimensional $x$ and a wide class of linear estimates. By considering the effect of bias we are able to relax assumptions on the class of functions $f$ which are considered. Simultaneous and numerical computations are used to illustrate the performance.

Nonparametric regression using locally weighted least squares was first discussed by Stone and by Cleveland. Recently, it was shown by Fan and by Fan and Gijbels that the local linear kernel-weighted least squares regression estimator has asymptotic properties making it superior, in certain senses, to the Nadaraya-Watson and Gasser-Muller kernel estimators. In this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables. We are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory. This approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors. We also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by Cleveland and Devlin and, in the univariate case, higher-order polynomial fits and derivative estimation.

Two results are presented concerning the consistency of the $k$-nearest neighbor regression estimate. We show that all modes of convergence in $L_1$ (in probability, almost sure, complete) are equivalent if the regression variable is bounded. Under the additional conditional $k/\log n \rightarrow \infty$ we also obtain the strong universal consistency of the estimate.

The paper is concerned with data from a collection of different, but related, regression curves $(m_j0_{j = 1}, \ldots, N, N \gg 1$. In statistical practice, analysis of such data is most frequently based on low-dimensional linear models. It is then assumed that each regression curve $m_j$ is a linear combination of a small number $L \ll N$ of common functions $g_1, \ldots, g_L$k. For example, if all $m_j$'s are straight lines, this holds with $L = 2, g_1 \equiv 1$ and $g_2(x) = x$. In this paper the assumption of a prespecified model is dropped. A nonparametric method is presented which allows estimation of the smallest $L$ and corresponding functions $g_1, \ldots, g_L$ from the data. The procedure combines smoothing techniques with ideas related to principal component analysis. An asymptotic theory is presented with yields detailed insight into properties of the resulting estimators. An application to household expenditure data illustrates the approach.

The $GI/G/1$ queueing model is regarded as a functional that maps the service and interarrival time distribution functions onto the stationary waiting time distribution function. By considering the output of the functional when it is applied to nonparametric estimators of the input distribution functions, we obtain a nonparametric estimator of the stationary waiting time distribution function. Using appropriate continuity and differentiability properties of the functional, we show that statistical properties of the input estimators carry over to corresponding properties for the stationary waiting time distribution function estimator.

Statistical tests consistent under any alternative are provided for both goodness-of-fit and two-sample comparisons. They are based on (1) the derivation of estimates for the tails of the probability distribution of some particular functionals of a Wiener sheet and (2) the use of a mapping that carries normalized empirical processes (asymptotically distributed as Brownian bridge) onto new processes that converge in distribution to a Wiener sheet. The proposed test statistics are easily obtained from the data. Hints for its computation and empirical estimations of the power of the tests are given.

If a criterion function $g(x_1, \ldots, x_m; \theta)$ depends on $m > 1$ samples, then a natural estimator of $\arg \max P^mg(x_1, \ldots, x_m; \theta)$ is the $\arg \max$ of a $U$-process. It is observed that, under suitable conditions, these estimators are asymptotically normal. This is then applied to prove asymptotic normality of Liu's simplical median and of Oja's medians in $\mathbb{R}^d$.

The influence function is determined for (twice) repeated median estimators with arbitrary kernel functions, and more generally in the case where the two medians are replaced by a general class of estimators. Asymptotic normality is then established for the repeated median estimator of the slope parameter in simple linear regression. In this case the influence function is bounded. For bivariate Gaussian data the efficiency becomes $4/\pi^2 \approx 40.5{\tt \%},$ which is the square of the efficiency of the univariate median. The asymptotic results are compared with finite-sample efficiencies. It turns out that the convergence to the asymptotic behavior is extremely slow.

Based on the duality between tests and confidence sets we introduce a new method to derive one-sided confidence bounds following the rejection of a null hypothesis with two-sided alternatives. This method imputes that the experimenter is only interested in confidence bounds if the null hypothesis is rejected. Furthermore, we suppose that he is only interested in the direction and a lower confidence bound concerning the distance of the true parameter value to the parameter values in the null hypothesis. If the null hypothesis is rejected, the new one-sided confidence bounds are always not worse than the corresponding bounds of the two-sided confidence interval approach. If the true parameter is far away from the null hypothesis, the new bounds tend to be nearly equal to the corresponding one-sided confidence bounds with full confidence level $1 - \alpha$. The new method will be studied and illustrated in more detail in one-parameter exponential families and location families with unimodal Lebesgue densities, and, as an example where conditional tests are available, we consider the comparison of two Poission distributions. In case of the normal distribution with unknown variance we propose among others a modification of a procedure of Hodges and Lehmann. Here it may be surprising that there exist situations where the new method yields confidence bounds exactly matching the classical one-sided confidence bounds.

The purpose of this paper is to give an explicit estimator dominating the positive-part James-Stein rule. The James-Stein estimator improves on the "usual" estimator $X$ of a multivariate normal mean vector $\theta$ if the dimension $p$ of the problem is at least 3. It has been known since at least 1964 that the positive-part version of this estimator improves on the James-Stein estimator. Brown's 1971 results imply that the positive-part version is itself inadmissible although this result was assumed to be true much earlier. Explicit improvements, however, have not previously been found; indeed, 1988 results of Bock and of Brown imply that no estimator dominating the positive-part estimator exists whose unbiased estimator of risk is uniformly smaller than that of the positive-part estimator.

Consider the model where $X_{ij}, i = 1, 2, \ldots, k, j = 1, 2, \ldots, n,$ are independent random variables distributed according to a one-parameter exponential family, with natural parameter $\theta_j$. We test $H_0: \theta_1 = \ldots = \theta_k$ versus $H_1: \theta \in \mathscr{C} - \{\theta: \theta \in H_0\},$ where $\theta = (\theta_1, \ldots, \theta_k)'$ and $\mathscr{C}$ is a cone determined by $A\theta \geq 0,$ where the rows of $A$ are contrasts with two nonzero elements. We offer a method of generating "good" tests for $H_0$ versus $H_1$. The method is to take a "good" test for $H_0$ versus $H_2:$ not $H_0,$ and apply the test to projected sample points, where the projection is onto $\mathscr{C}$. "Good" tests for $H_0$ versus $H_2$ are tests that are Schur convex. "Good" tests for $H_0$ versus $H_1$ are tests which are monotone with respect to a cone order. We demonstrate that if the test function for $H_0$ versus $H_2$ is a constant size Schur convex test, then the resulting projected test is monotone.

Anderson studied the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric convex set. Restricting attention to elliptically contoured, unimodal densities, Mukerjee, Robertson and Wright weakened the assumption of symmetry on the set and obtained monotonicity properties of power functions, including unbiasedness, for some likelihood ratio tests in order restricted inference for the variance-known case. For elliptically contoured, unimodal densities, we weaken the assumption of convexity to obtain similar results in the case of unknown variances. The results apply to situations in which the null hypothesis is a linear space and the alternative is a closed, convex cone.

Let $\mathscr{K} = \{K_\lambda: \lambda \in \Lambda\}$ be a family of sampling distributions for the data $x$ on a sample space $\mathscr{X}$ which is indexed by a parameter $\lambda \in \Lambda,$ and let $\mathscr{F}$ be a family of priors on $\Lambda$. Then $\mathscr{F}$ is said to be conjugate for $\mathscr{K}$ if it is closed under sampling, that is, if the posterior distributions of $\lambda$ given the data $x$ belong to $\mathscr{F}$ for almost all $x$. In this paper, we set up a framework for the study of what we term the dual problem: for a given family of priors $\mathscr{F}$ (a subfamily of a general exponential family), find the class of sampling models $\mathscr{K}$ for which $\mathscr{F}$ is conjugate. In particular, we show that $\mathscr{K}$ must be a general exponential family dominated by some measure $Q$ on $(\mathscr{X}, B),$ where $B$ is the Borel field on $\mathscr{X}$. It is the class of such measures $Q$ that we investigate in this paper. We study its geometric features and general structure and apply the results to some familiar examples.

A new preferred point geometric structure for statistical analysis, closely related to Amari's $\alpha$-geometries, is introduced. The added preferred point structure is seen to resolve the problem that divergence measures do not obey the intutively natural axioms for a distance function as commonly used in geometry. Using this tool, two key results of Amari which connect geodesics and divergence functions are developed. The embedding properties of the Kullback-Leibler divergence are considered and a strong curvature condition is produced under which it agrees with a statistically natural (squared) preferred point geodesic distance. When this condition fails the choice of divergence may be crucial. Further, Amari's Pythagorean result is shown to generalise in the preferred point context.

We construct a statistical test for a sequence alignment problem which enables us to decide whether two given sequences are related. Such a test can be used in DNA and protein sequence comparisons. It is based on a comparison of two long sequences of i.i.d. letters taken from a finite alphabet. The test statistic typically employed is the length of the longest matching region between the two sequences in which a certain number of insertions and deletions but no mismatches are allowed. We give a distributional result which enables one to compute $P$-values, and hence to decide whether or not the two sequences are related. Its proof utilizes the Chen-Stein method for Poisson approximation. The test is based on a greedy algorithm that searches for the longest matching region. We show that this algorithm finds the longest matching region with probability approaching 1 as the lengths of the two sequences go to infinity.

Comparison of some normal experiments involving intraclass and inter-class correlation is reduced to comparison of one-way random normal experiments. In consequence, the main result in Shaked and Tong is generalized and completed.