Noël Veraverbeke

Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing

We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445-464], Fermanian, Radulovic and Wegkamp [Bernoulli 10 (2004) 847-860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265-282]. All of these estimators have as a drawback that they can suffer from a comer bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias problem. For Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445-464] and Chen and Huang [Canad. J. Statist. 35 (2007) 265-282], the improvement involves shrinking the bandwidth with an appropriate functional factor; for Fermanian, Radulovic and Wegkamp [Bernoulli 10 (2004) 847-860], this is done by using a transformation. The theoretical contribution of the paper is a weak convergence result for the three improved estimators under conditions that are met for most copula families. We also discuss the choice of bandwidth parameters, theoretically and practically, and illustrate the finite-sample behaviour of the estimators in a simulation study. The improved estimators are applied to goodness-of-fit testing for copulas.

Estimation and Bootstrap with Censored Data in Fixed Design Nonparametric Regression

We study Berans extension of the Kaplan-Meier estimator for thesituation of right censored observations at fixed covariate values. Thisestimator for the conditional distribution function at a given value of thecovariate involves smoothing with Gasser-Müller weights. We establishan almost sure asymptotic representation which provides a key tool forobtaining central limit results. To avoid complicated estimation ofasymptotic bias and variance parameters, we propose a resampling methodwhich takes the covariate information into account. An asymptoticrepresentation for the bootstrapped estimator is proved and the strongconsistency of the bootstrap approximation to the conditional distributionfunction is obtained.

Change-point problem and bootstrap

We consider a class of simple estimators of the change-point m in a sequence of n independent random variables X1…,X n satisfying E(X i ) = θ0 for i = 1,…,m and E(X i ) = θ0+δ n for i = m +1,…n. (θ0 and δ n are unknown). We obtain rates of consistency for the estimator, derive its limiting distribution and show that the bootstrap approximation is asymptotically valid. The results are illustrated by some simulations.

RESAMPLING FROM CENTERED DATA IN THE TWO-SAMPLE PROBLEM*

Abstract Bootstrap and permutation approximations to the distribution of U-statistics are shown to be valid when the resampling is from residuals in the two-sample problem. The motivation for using residuals comes from testing for homogeneity of scale in the presence of nuisance location parameters. New asymptotic results for U-statistics with estimated parameters are key tools in the proofs.

Presmoothed Kaplan–Meier and Nelson–Aalen estimators

In this article, a modification of the Kaplan–Meier and Nelson–Aalen estimators in the right random censorship model is studied. The new estimators are obtained by replacing the censoring indicator variables in the classical definitions by values of a nonparametric regression estimator. Asymptotic normality is obtained and it is shown that this presmoothing idea leads to a gain in asymptotic mean squared error. A local plug-in bandwidth selector is introduced and the problem of optimal pilot bandwidth selection for this estimator is studied. The gain of the presmoothed estimators with automatic plug-in bandwidth selector is demonstrated in a simulation study.

Estimation of the conditional distribution in regression with censored data: a comparative study

In nonparametric regression with censored data, the conditional distribution of the response given the covariate is usually estimated by the Beran (Technical Report, University of California, Berkeley, 1981) estimator. This estimator, however, is inconsistent in the right tail of the distribution when heavy censoring is present. In an attempt to solve this inconsistency problem of the Beran estimator, Van Keilegom and Akritas (Ann. Statist. (1999)) developed an alternative estimator for heteroscedastic regression models (see (1.1) below for the definition of the model), which behaves well in the right tail even under heavy censoring. In this paper, the finite sample performance of the estimator introduced by Van Keilegom and Akritas (Ann. Statist. (1999)) and the Beran (Technical Report, University of California, Berkeley, 1981) estimator is compared by means of a simulation study. The simulations show that both the bias and the variance of the former estimator are smaller than that of the latter one. Also, these estimators are used to analyze the Stanford heart transplant data.

Bootstrapping quantiles in a fixed design regression model with censored data

We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation.

Semiparametric estimation of conditional copulas

The manner in which two random variables influence one another often depends on covariates. A way to model this dependence is via a conditional copula function. This paper contributes to the study of semiparametric estimation of conditional copulas by starting from a parametric copula function in which the parameter varies with a covariate, and leaving the marginals unspecified. Consequently, the unknown parts in the model are the parameter function and the unknown marginals. The authors use a local pseudo-likelihood with nonparametrically estimated marginals approximating the unknown parameter function locally by a polynomial. Under this general setting, they prove the consistency of the estimators of the parameter function as well as its derivatives; they also establish asymptotic normality. Furthermore, they derive an expression for the theoretical optimal bandwidth and discuss practical bandwidth selection. They illustrate the performance of the estimation procedure with data-driven bandwidth selection via a simulation study and a real-data case.

Hazard rate estimation in nonparametric regression with censored data

Consider a regression model in which the responses are subject to random right censoring. In this model, Beran studied the nonparametric estimation of the conditional cumulative hazard function and the corresponding cumulative distribution function. The main idea is to use smoothing in the covariates. Here we study asymptotic properties of the corresponding hazard function estimator obtained by convolution smoothing of Berans cumulative hazard estimator. We establish asymptotic expressions for the bias and the variance of the estimator, which together with an asymptotic representation lead to a weak convergence result. Also, the uniform strong consistency of the estimator is obtained.

Weak convergence of the bootstrapped conditional kaplan-meier process and its quantile process

We consider a fixed design regression model in which the observations are subject to random right censoring. We establish weak convergence results for Berans conditional Kaplan-Meier estimator and the corresponding quantile estimator, as well as for their bootstrapped versions. This enables us to construct bootstrap confidence bands for both the distribution and the quantile function.