Let $Y_1,\ldots, Y_n$ be independent identically distributed with density $p_0$ and let $\mathscr{F}$ be a space of densities. We show that the supremum of the likelihood ratios $\prod^n_{i=1} p(Y_i)/p_0(Y_i)$, where the supremum is over $p \in \mathscr{F}$ with $\|p^{1/2} - p^{1/2}_0\|_2 \geq \varepsilon$, is exponentially small with probability exponentially close to 1. The exponent is proportional to $n\varepsilon^2$. The only condition required for this to hold is that $\varepsilon$ exceeds a value determined by the bracketing Hellinger entropy of $\mathscr{F}$. A similar inequality also holds if we replace $\mathscr{F}$ by $\mathscr{F}_n$ and $p_0$ by $q_n$, where $q_n$ is an approximation to $p_0$ in a suitable sense. These results are applied to establish rates of convergence of sieve MLEs. Furthermore, weak conditions are given under which the "optimal" rate $\varepsilon_n$ defined by $H(\varepsilon_n, \mathscr{F}) = n\varepsilon^2_n$, where $H(\cdot, \mathscr{F})$ is the Hellinger entropy of $\mathscr{F}$, is nearly achievable by sieve estimators.

We derive upper bounds for the coverage error of confidence intervals for a population mean uniformly over large classes of populations and different types of confidence intervals. It is shown that the order of these bounds is achieved by the normal approximation method for constructing confidence intervals, uniformly over distributions with finite third moment, and, by an empirical Edgeworth correction of this approach, uniformly over smooth distributions with finite fourth moments. These results have straightforward extensions to higher orders of Edgeworth correction and higher orders of moments. Our upper bounds to coverage accuracy are based on Berry-Esseen theorems for Edgeworth expansions of the distribution of the Studentized mean.

This paper considers efficient estimation of the Euclidean parameter $\theta$ in the proportional odds model $G(1 - G)^{-1} = \theta F(1 - F)^{-1}$ when two independent i.i.d. samples with distributions $F$ and $G$, respectively, are observed. The Fisher information $I(\theta)$ is calculated based on the solution of a pair of integral equations which are derived from a class of more general semiparametric models. A one-step estimate is constructed using an initial $\sqrt N$-consistent estimate and shown to be asymptotically efficient in the sense that its asymptotic risk achieves the corresponding minimax lower bound.

This paper introduces the concept of dual likelihood as a method of improving accuracy in inference situations depending on martingale estimating equations. Asymptotic results are given for the dual likelihood ratio statistic, and the structure of the family of alternatives is explored. Applications to survival analysis and also to time series, likelihood inference and independent observations are given. Connections to nonparametric likelihood (including empirical likelihood) are established.

Let $\hat{F}_n$ be the Kaplan-Meier estimator of a distribution function $F$ computed from randomly censored data. We show that under optimal integrability assumptions on a function $\varphi$, the Kaplan-Meier integral $\int \varphi d\hat{F}_n$, when properly standardized, is asymptotically normal.

In this paper, we prove that the Lynden-Bell estimator of a distribution function in the random truncation model is uniformly strong consistent over the whole half line, a problem left open by Woodroofe.

It has been over 60 years since Kolmogorov introduced a distribution-free omnibus test for the simple null hypothesis that a distribution function coincides with a given distribution function. Doob subsequently observed that Kolmogorov's approach could be simplified by transforming the empirical process to an empirical process based on uniform random variables. Recent use of more sophisticated transformations has led to the construction of asymptotically distribution-free omnibus tests when unknown parameters are present. The purpose of the present paper is to use the transformation approach to construct an asymptotically distribution-free omnibus test for independence of a survival time from a covariate. The test statistic is obtained from a certain test statistic process (indexed by time and covariate), which is shown to converge in distribution to a Brownian sheet. A simulation study is carried out to investigate the finite sample properties of the proposed test and an application to data from the British Medical Research Council's 4th myelomatosis trial is given.

Let $g: \lbrack 0, 1\rbrack \rightarrow \lbrack 0, 1\rbrack$ be a monotone nondecreasing function and let $G$ be the closure of the set $\{(x, y) \in \lbrack 0, 1\rbrack \times \lbrack 0, 1\rbrack: 0 \leq y \leq g (x)\}$. We consider the problem of estimating the set $G$ from a sample of i.i.d. observations uniformly distributed in $G$. The estimation error is measured in the Hausdorff metric. We propose the estimator which is asymptotically efficient in the minimax sense.

Let $n > p > k > 0$ be integers. Let $\delta$ be any technique for fitting $k$-planes to $p$-variate data sets of size $n$, for example, linear regression, principal components or projection pursuit. Let $\mathscr{Y}$ be the set of data sets which are (1) singularities of $\delta$, that is, near them $\delta$ is unstable (for example, collinear data sets are singularities of least squares regression) and (2) nondegenerate, that is, their rank, after centering, is at least $k$. It is shown that the Hausdorff dimension, $\dim_H(\mathscr{Y})$, of $\mathscr{Y}$ is at least $nk + (k + 1)(p - k) - 1$. This bound is tight. Under hypotheses satisfied by some projection pursuits (including principal components), $\dim_H(\mathscr{Y}) \geq np - 2$, that is, once singularity is taken into account, only two degrees of freedom remain in the problem! These results have implications for multivariate data description, resistant plane-fitting and jackknifing and bootstrapping plane-fitting.

In this paper optimal rates of convergence are derived for estimates of sets in $N$-dimensional "black and white" pictures under smoothness conditions. It is assumed that the boundaries of the "black" regions have a smooth parameterisation, that is, that the boundaries are given by smooth functions from the sphere $S^{N-1}$ into $\mathbb{R}^N$. Furthermore, classes of convex regions are considered. Two models are studied: edge estimation models motivated by image segmentation problems and density support estimation.

A conservative finite-sample simultaneous confidence envelope for a density can be found by solving a finite set of finite-dimensional linear programming problems if the density is known to be monotonic or to have at most $k$ modes relative to a positive weight function. The dimension of the problems is at most $(n/\log n)^{1/3}$, where $n$ is the number of observations. The linear programs find densities attaining the largest and smallest values at a point among cumulative distribution functions in a confidence set defined using the assumed shape restriction and differences between the empirical cumulative distribution function evaluated at a subset of the observed points. Bounds at any finite set of points can be extrapolated conservatively using the shape restriction. The optima are attained by densities piecewise proportional to the weight function with discontinuities at a subset of the observations and at most five other points. If the weight function is constant and the density satisfies a local Lipschitz condition with exponent $\varrho$, the width of the bounds converges to zero at the optimal rate $(\log n/n)^{\varrho/(1+2\varrho)}$ outside every neighborhood of the set of modes, if a "bandwidth" parameter is chosen correctly. The integrated width of the bounds converges at the same rate on intervals where the density satisfies a Lipschitz condition if the intervals are strictly within the support of the density. The approach also gives algorithms to compute confidence intervals for the support of monotonic densities and for the mode of unimodal densities, lower confidence intervals on the number of modes of a distribution and conservative tests of the hypothesis of $k$-modality. We use the method to compute confidence bounds for the probability density of aftershocks of the 1984 Morgan Hill, CA, earthquake, assuming aftershock times are an inhomogeneous Poisson point process with decreasing intensity.

Given data from a sample of noisy curves, we consider a nonlinear parametric regression model with unknown model function. An iterative algorithm for estimating individual parameters as well as the model function is introduced under the assumption of a certain shape invariance: the individual regression curves are obtained from a common shape function by linear transformations of the axes. Our algorithm is based on least-squares methods for parameter estimation and on nonparametric kernel methods for curve estimation. Asymptotic distributions are derived for the individual parameter estimators as well as for the estimator of the shape function. An application to human growth data illustrates the method.

This paper is concerned with estimating the loss of a point estimator when sampling from a spherically symmetric distribution. We examine the canonical setting of a general linear model where the dimension of the parameter space is greater than 4 and less than the dimension of the sampling space. We consider two location estimators--the least squares estimator and a shrinkage estimator--and we compare their unbiased loss estimator with an improved loss estimator. The domination results are valid for a large class of spherically symmetric distributions and, in so far as the sampling distribution does not need to be precisely specified, the estimates have desirable robustness properties.

The estimation of a common mean vector $\theta$ given two independent normal observations $X \sim N_p(\theta, \sigma^2_x I)$ and $Y \sim N_p(\theta, \sigma^2_y I)$ is reconsidered. It being known that the estimator $\eta X + (1 - \eta)Y$ is inadmissible when $\eta \in (0, 1)$, we show that when $\eta$ is 0 or 1, then the opposite is true, that is, the estimator is admissible. The general situation is that an estimator $X^\ast$ can be improved by shrinkage when there exists a statistic $B$ which, in a certain sense, estimates a lower bound on the risk of $X^\ast$. On the other hand, an estimator is admissible under very general conditions if there is no reasonable way to detect that its risk is small.

Suppose that $X_1,\ldots, X_m$ are i.i.d. from a continuous distribution function $F$ and $Y_1,\ldots, Y_n$ are i.i.d. from a continuous distribution function $G; X$'s and $Y$'s are independent. A minimum variance unbiased estimator of $P\{X < Y\}$ is the Mann-Whitney statistic. We show that the Mann-Whitney statistic is admissible under a class of weighted squared error losses and is minimax under a proper weighted squared error loss.

We suppose that our observations can be decomposed into a fixed signal plus random noise, where the noise is modelled as a particular stationary Gaussian random field in $N$-dimensional Euclidean space. The signal has the form of a known function centered at an unknown location and multiplied by an unknown amplitude, and we are primarily interested in a test to detect such a signal. There are many examples where the signal scale or width is assumed known, and the test is based on maximising a Gaussian random field over all locations in a subset of $N$-dimensional Euclidean space. The novel feature of this work is that the width of the signal is also unknown and the test is based on maximising a Gaussian random field in $N + 1$ dimensions, $N$ dimensions for the location plus one dimension for the width. Two convergent approaches are used to approximate the null distribution: one based on the method of Knowles and Siegmund, which uses a version of Weyl's tube formula for manifolds with boundaries, and the other based on some recent work by Worsley, which uses the Hadwiger characteristic of excursion sets as introduced by Adler. Finally we compare the power of our method with one based on a fixed but perhaps incorrect signal width.

Certain three-dimensional images arising in medicine and astrophysics are modelled as a smooth random field, and experimenters are interested in the number of peaks or "hot-spots" present in such an image. This paper studies the Hadwiger characteristic of the excursion set of a random field. The excursion set is the set of points where the image exceeds a fixed threshold, and the Hadwiger characteristic, like the Euler characteristic, counts the number of connected components in the excursion set minus the number of "holes." For high thresholds the Hadwiger characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler, who defined the IG (integral geometry) characteristic of excursion sets as a multidimensional analogue of the number of "upcrossings" of the threshold by a unidimensional process. The IG characteristic equals the Euler characteristic of an excursion set provided that the set does not touch the boundary of the volume, and Adler found the expected IG characteristic for a stationary random field inside a fixed volume. Worsley, Evans, Marrett and Neelin used the IG characteristic as an estimator of the number of regions of activation of positron emission tomography (PET) images of blood flow in the brain. Unfortunately the IG characteristic is only defined on intervals: it is not invariant under rotations and it can fail to count connected regions that touch the boundary. This is important since activation often occurs in the cortical regions near the boundary of the brain. In this paper we study the Hadwiger characteristic, which is defined on arbitrary sets, is invariant under rotations and does count connected regions whether they touch the boundary or not. Our main result is a simple expression for the expected Hadwiger characteristic for an isotropic stationary random field in two and three dimensions and on a smooth surface embedded in three dimensions. Results are applied to PET studies of pain perception and word recognition.

This paper develops extensions of the regression quantiles of Koenker and Bassett (1978) to autoregression. It generalizes several results of Jureckova (1992a) and Gutenbrunner and Jureckova (1992) in linear regression to autoregression models. In particular, it gives the asymptotic uniform linearity of linear rank-scores statistics based on residuals suitable in autoregression. It also discusses the two types of $L$-statistics appropriate in autoregression.

A method is proposed to deal with the problem of blind deconvolution of a special non-Gaussian linear process, in which the input to the linear system is a real- or complex-valued multilevel random sequence that satisfies certain regularity conditions. The gist of the method is to apply a linear filter to the observed process and adjust the filter until a multilevel output is obtained. It is shown that the deconvolution problem can be solved (with only scale/rotation and shift ambiguities) if the output sequence of the filter contains a subsequence that converges weakly to a multilevel random variable. A cost function is proposed so that any minimizer of the cost function provides a solution to the deconvolution problem. Moreover, when the process is parametric, it is shown that a consistent estimator of the parameter can be obtained by minimizing an empirical criterion. The estimation accuracy is shown to depend on the tail behavior of the inverse system, which in many cases decays exponentially as the sample size grows. Special consideration is given to applications in the equalization of digital communications systems.