Let $\{P_{\vartheta,\eta}:(\vartheta, \eta) \in \Theta \times H\}$, with $\Theta \subset \mathbb{R}$ and H arbitrary, be a family of mutually absolutely continuous probability measures on a measurable space $(X, \mathscr{A})$. The problem is to estimate $\vartheta$, based on a sample $(x_1, \cdots, x_n)$ from $\times^n_1 P_{\vartheta,\eta_\nu}$. If $(\eta_1, \cdots, \eta_n)$ are independently distributed according to some unknown prior distribution $\Gamma$, then the distribution of $n^{1/2}(\vartheta^{(n)} - \vartheta)$ under $P^n_{\vartheta,\Gamma}(P_{\vartheta, \Gamma}$ being the $\Gamma$-mixture of $P_{\vartheta,\eta}, \eta \in H$) cannot be more concentrated asymptotically than a certain normal distribution with mean 0, say $N_{(0, \sigma^2_0(\vartheta,\Gamma))}$. Folklore says that such a bound is also valid if $(\eta_1, \cdots, \eta_n)$ are just unknown values of the nuisance parameter: In this case, the distribution cannot be more concentrated asymptotically than $N_{(0, \sigma^2_0(\vartheta,E^{(n)}_{(\eta_1, \cdots, \eta_n)}))}$, where $E^{(n)}_{(\eta_1, \cdots, \eta_n)}$ is the empirical distribution of $(\eta_1,\cdots, \eta_n)$. The purpose of the present paper is to discuss to which extent this conjecture is true. The results are summarized at the end of Sections 1 and 3.

A Berry-Esseen theorem for the rate of convergence of general nonlinear multivariate sampling statistics with normal limit distribution is derived via a multivariate extension of Stein's method. The result generalizes in particular previous results of Bolthausen for one-dimensional linear rank statistics, one-dimensional results of van Zwet and Friedrich for general functions of independent random elements and provides convergence bounds for general multivariate sampling statistics without restrictions on the sampling proportions.

A simple and tractable iterative least squares estimation procedure for censored regression models with known error distributions is analyzed. It is found to be equivalent to a well-defined Huber type $M$-estimate. Under a regularity condition, the algorithm converges geometrically to a unique solution. The resulting estimate is $\sqrt N$-consistent and asymptotically normal.

New estimators for Cox regression are considered. Their asymptotic properties, both on and off the model, are established. Corollaries include conditions under which the maximum partial likelihood estimator defines a parameter in the population and the asymptotics of the case-cohort estimator. Robust estimators that minimize the asymptotic variance subject to a bound on the maximal bias on infinitesimal neighborhoods are discussed. The estimators are illustrated with medical data.

For the two-sample problem with randomly censored data, there exists a general asymptotic theory of rank statistics which are functionals of stochastic integrals with respect to certain empirical martingales. In the present paper a conditional counterpart of this theory is developed. The conditional martingales are versions of the original ones reduced to the unit interval having their jumps at fixed lattice points. The resulting conditional tests are strictly distribution free under the null hypothesis of randomness if the censoring distributions in both samples are equal and are asymptotically equivalent to their unconditional counterparts even if the censoring distributions are different. Simulations for linear rank statistics and Kolmogorov-Smirnov-type statistics show superiority of the conditional versions over their unconditional counterparts with respect to size and robustness under unequal censoring in both samples. At the same time the power of the conditional and unconditional tests is very similar in most cases.

We suggest bootstrap methods for constructing confidence bands (and intervals) for an unknown linear functional relationship in an errors-invariables model. It is assumed that the ratio of error variances is known to lie within an interval $\Lambda = \lbrack\lambda_1, \lambda_2\rbrack$. A confidence band is constructed for the range of possible linear relationships when $\lambda \in \Lambda$. Meaningful results are obtained even in the extreme case $\Lambda = \lbrack 0, \infty\rbrack$, which corresponds to no assumption being made about $\Lambda$. The bootstrap bands have several interesting features, which include the following: (i) the bands do not shrink to a line as $n \rightarrow \infty$, unless $\Lambda$ is a singleton (i.e., $\lambda_1 = \lambda_2)$; (ii) percentile-$t$ versions of the bands enjoy only first-order coverage accuracy, not the second-order accuracy normally found in simpler statistical problems.

It is known that the least squares cross-validation bandwidth is asymptotically optimal in the case of kernel-based density and hazard rate estimation in the settings of both complete and randomly right-censored samples. From a practical point of view, it is important to know at what rate the cross-validation bandwidth converges to the optimal. In this paper we answer this question in a general setup which unifies all four possible cases.

A generalized bootstrap version is defined for degenerate $U$-statistics. Our main result shows that the (conditional) distribution function of the bootstrapped degenerate $U$-statistic provides a consistent estimator for the unknown distribution function of the degenerate $U$-statistic under consideration. For the proof we rely on a rank statistic approach.

A natural measure of the degree of robustness of an estimate $\mathbf{T}$ is the maximum asymptotic bias $B_\mathbf{T}(\varepsilon)$ over an $\varepsilon$-contamination neighborhood. Martin, Yohai and Zamar have shown that the class of least $\alpha$-quantile regression estimates is minimax bias in the class of $M$-estimates, that is, they minimize $B_\mathbf{T}(\varepsilon)$, with $\alpha$ depending on $\varepsilon$. In this paper we generalize this result, proving that the least $\alpha$-quantile estimates are minimax bias in a much broader class of estimates which we call residual admissible and which includes most of the known robust estimates defined as a function of the regression residuals (e.g., least median of squares, least trimmed of squares, $S$-estimates, $\tau$-estimates, $M$-estimates, signed $R$-estimates, etc.). The minimax results obtained here, likewise the results obtained by Martin, Yohai and Zamar, require that the carriers have elliptical distribution under the central model.

Section 1 of the paper contains a general discussion of robustness. In Section 2 the influence function of the Hampel-Rousseeuw least median of squares estimator is derived. Linearly invariant weak metrics are constructed in Section 3. It is shown in Section 4 that $S$-estimators satisfy an exact Holder condition of order 1/2 at models with normal errors. In Section 5 the breakdown points of the Hampel-Krasker dispersion and regression functionals are shown to be 0. The exact breakdown point of the Krasker-Welsch dispersion functional is obtained as well as bounds for the corresponding regression functional. Section 6 contains the construction of a linearly equivariant, high breakdown and locally Lipschitz dispersion functional for any design distribution. In Section 7 it is shown that there is no inherent contradiction between efficiency and a high breakdown point. Section 8 contains a linearly equivariant, high breakdown regression functional which is Lipschitz continuous at models with normal errors.

The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction of a new class of kernel estimators. It is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution. In fact, there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth. It is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate, which is rich enough for many practical applications. Furthermore, to achieve optimality, we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators.

In general, there will be visible differences between a parametric and a nonparametric curve estimate. It is therefore quite natural to compare these in order to decide whether the parametric model could be justified. An asymptotic quantification is the distribution of the integrated squared difference between these curves. We show that the standard way of bootstrapping this statistic fails. We use and analyse a different form of bootstrapping for this task. We call this method the wild bootstrap and apply it to fitting Engel curves in expenditure data analysis.

Using an expression for the expected number of local maxima of a random field, we derive an upper bound for the volume of a tube about a manifold in the unit sphere and show that under certain conditions our bound agrees with the evaluation of the tube volume in Weyl's formula. Applications to tests and confidence regions in nonlinear regression are discussed.

Consider the problem of constructing a prediction region $D_n$ for a potentially observable variable $X$ on the basis of a learning sample of size $n$. Usually, the requirement that $D_n$ contain $X$ with probability $\alpha$, conditionally on the learning sample, does not uniquely determine $D_n$. This paper develops a general probability-centering concept for prediction regions that extends to vector-valued or function-valued $X$ the classical notion of an equal-tailed prediction interval. The dual requirements of probability centering and specified coverage probability determine $D_n$ uniquely. Several examples illustrate the scope and consequences of the proposed centering concept.

Let $X_1, X_2, \cdots, X_n$ be a random sample from a continuous univariate distribution $F$, and let $\Delta = \prod_{1 \leq i < j \leq n}(X_i - X_j)^2$ denote the discriminant, or square of the Vandermonde determinant, constructed from the random sample. The statistic $\Delta$ arises in the study of moment matrices and inference for mixture distributions, the spectral theory of random matrices, control theory and statistical physics. In this paper, we study the probability distribution of $\Delta$. When $X_1, \cdots, X_n$ is a random sample from a normal, gamma or beta population, we use Selberg's beta integral formula to obtain stochastic representations for the exact distribution of $\Delta$. Further, we obtain stochastic bounds for $\Delta$ in the normal and gamma cases. Using the theory of $U$-statistics, we derive the asymptotic distribution of $\Delta$ under certain conditions on $F$.

This article develops an axiomatic basis for the relationship between conditional independence and graphical models in statistical analysis. In particular, the following relationships are established: (1) every axiom for conditional independence is an axiom for graph separation, (2) every graph represents a consistent set of independence and dependence constraints, (3) all binary factorizations of strictly positive probability models can be encoded and determined in polynomial time using their correspondence to graph separation, (4) binary factorizations of non-strictly positive probability models can also be derived in polynomial time albeit less efficiently and (5) unconditional independence relative to normal models can be axiomatized with a finite set of axioms.

We consider the estimation problem of the parameter $b$ of a stationary AR($p$) process without any of the usual causality assumptions. The aim of the paper is to derive asymptotic minimax bounds for estimators of $b$. When the distribution of the noise is known, we show LAN properties of the model and derive locally asymptotically minimax (LAM) estimators. The most important results are about the case of unknown distribution: The main result shows that, if one uses the usual parametrization, these bounds depend heavily on the causality or the noncausality of the process, so that adaptive efficient estimation is impossible in the noncausal situation: The scaling factor is shown to give the hardest one-dimensional subproblem, and an unusual scaling is exhibited that could lead to adaptive efficient estimation of the rescaled parameter even in the noncausal case.

Under stationarity and weak dependence, the statistical significance and the estimation of the extremal index are considered. It is shown that the distribution of the sample maximum can be uniformly approximated given the extremal index and the marginal distribution as the sample size increases. An adaptive procedure is proposed for estimating the extremal index. The procedure is shown to be asymptotically optimal in a class of estimators.

Coster and Cheng presented a generalized foldover scheme for the construction of systematic run orders of fractional factorial designs, with all factors having the same prime power number of levels, for which all the main effects components of the factors are orthogonal to a polynomial trend present in every block of the design. In this paper, we present modifications to the foldover method that allow polynomial trend-free run orders to be constructed in the following more general settings: Designs for which the number of levels of each factor is not a prime power; mixed-level factorial designs with factors at different numbers of levels; cases in which some or all two- and higher-factor interactions, not just the main effects, are required to be orthogonal to the polynomial trend.

Yeh and Bradley conjectured that every binary connected block design with blocks of size $k$ and a constant replication number $r$ for each treatment can be converted to a linear trend-free design by permuting the positions of treatments within blocks if and only if $r(k + 1) \equiv 0 (\operatorname{mod} 2)$. This conjecture is studied. Results include: (i) the conjecture is true whenever the block size is even and (ii) the conjecture is true for BIB designs.

A response-adaptive treatment allocation design for a clinical trial attempts to place the majority of patients on the treatment that appears more successful, based on the responses of patients already treated. One example of such a design is the randomized play-the-winner rule developed by Wei and Durham, which randomizes the treatment assignment probabilities according to the outcomes of treatments previously assigned. For a trial with dichotomous treatment responses and a randomized play-the-winner assignment scheme, exact small sample permutation tests of the hypothesis of equal treatment effects and large sample tests based on a population model have previously been developed. We present a large sample permutation test statistic for this case; under certain conditions on the sequence of responses, the test statistic is shown to be asymptotically normal. For a trial with a continuous response variable, we develop a rank-based analog of the randomized play-the-winner assignment scheme. Simulation evidence in both cases suggests that a normal approximation to the test statistic works well for moderate-sized trials, with some conservatism in the extreme tails.

The performance of Bayes estimates are studied, under an assumption of conditional exchangeability. More exactly, for each subject in a data set, let $\xi$ be a vector of binary covariates and let $\eta$ be a binary response variable, with $P\{\eta = 1\mid \xi\} = f(\xi)$. Here, $f$ is an unknown function to be estimated from the data; the subjects are independent, and satisfy a natural "balance" condition. Define a prior distribution on $f$ as $\sum_kw_k\pi_k/\sum_kw_k$, where $\pi_k$ is uniform on the set of $f$ which only depend on the first $k$ covariates and $w_k > 0$ for infinitely many $k$. Bayes estimates are consistent at all $f$ if $w_k$ decreases rapidly as $k$ increase. Otherwise, the estimates are inconsistent at $f \equiv 1/2$.

Bayesian statistical inference for sampling from weighted distribution models is studied. Small-sample Bayesian bootstrap clone (BBC) approximations to the posterior distribution are discussed. A second-order property for the BBC in unweighted i.i.d. sampling is given. A consequence is that BBC approximations to a posterior distribution of the mean and to the sampling distribution of the sample average, can be made asymptotically accurate by a proper choice of the random variables that generate the clones. It also follows from this result that in weighted sampling models, BBC approximations to a posterior distribution of the reciprocal of the weighted mean are asymptotically accurate; BBC approximations to a sampling distribution of the reciprocal of the empirical weighted mean are also asymptotically accurate.

Minimal sufficiency and completeness are examined for the multistage, multihit and Weibull quantal response models. It is shown that the response counts are minimal sufficient statistics and conditions are presented for completeness for the families of these models. These results provide an example of a complete sufficient statistic for a curved exponential family which is of higher dimension than the parameter space. Uniformly minimum variance unbiased (UMVU) estimators may not exist for the probability of response at a given dose if the response counts are not complete sufficient statistics.

A general result concerning noncompleteness of location families of probability measures on Euclidean space is pointed out. Examples include boundedly complete families, such as those generated by certain scale mixtures of the standard Gaussian distribution. These examples illuminate completeness criteria for location families and compare favourably in simplicity with previously known examples of incomplete boundedly complete (nonlocation) families.

Let $X_1, X_2, \cdots$ be a sequence of random variables, $(X_1, \cdots, X_n) \sim F^n_\theta, \theta \in \Theta$. In a work by Bahadur it was shown that, for some sequential problems, an inference may be based on a sequence of sufficient and transitive statistics $S_n = S_n(X_1, \cdots, X_n)$ without any loss in statistical performance. A simple criterion for transitivity is given in Theorem 1.