# I. GijbelsKU Leuven | ku leuven · Department of Mathematics

I. Gijbels

Professor

## About

166

Publications

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9,117

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Citations since 2017

## Publications

Publications (166)

Multivariate tail coefficients are an important tool when investigating dependencies between extreme events for different components of a random vector. Although bivariate tail coefficients are well-studied, this is, to a lesser extent, the case for multivariate tail coefficients. This paper contributes to this research area by (i) providing a thor...

We introduce a new class of robust M-estimators for performing simultaneous parameter estimation and variable selection in high-dimensional regression models. We first explain the motivations for the key ingredient of our procedures which are inspired by regularization methods used in wavelet thresholding in noisy signal processing. The derived pen...

Allowing for symmetry in distributions is often a necessity in statistical modelling. This paper studies a broad family of asymmetric densities, which in a regression setting shares basic philosophy with generalized (non)linear models. The main focus, however, for the family of densities studied here is quantile estimation instead of mean estimatio...

In this paper, we provide a detailed study of a general family of asymmetric densities. In the general framework, we establish expressions for important characteristics of the distributions and discuss estimation of the parameters via method‐of‐moments as well as maximum likelihood estimation. Asymptotic normality results for the estimators are pro...

The paper concerns robust estimation and variable selection in heteroscedastic linear regression models. After a brief review of existing methods for estimation in such models, a robust S-estimation approach is discussed. For all methods concise descriptions of algorithms are provided. Little is available upon robust variable selection methods for...

Automated or data-driven bandwidth selection methods tend to break down in the presence of correlated errors. While this problem has previously been studied in the fixed design setting for kernel regression, the results were applicable only when there is knowledge about the correlation structure. This article generalizes these results to the random...

We provide a computational framework for the selection of weights (ω1,…,ωd) that minimize the expected shortfall of the aggregated risk Z=∑i=1dωiXi. Contrary to classic and recent results, we neither restrict the marginal distributions nor the dependence structure of (X1,…,Xd) to any specific type. While the margins can be set to any absolutely con...

Quantiles and expectiles of a distribution are found to be useful descriptors of its tail in the same way as the median and mean are related to its central behavior. This paper considers a valuable alternative class to expectiles, called extremiles, which parallels the class of quantiles and includes the family of expected minima and expected maxim...

en In mean regression the characteristic of interest is the conditional mean of the response given the covariates. In quantile regression the aim is to estimate any quantile of the conditional distribution function. For given covariates, the conditional quantile function fully characterizes the entire conditional distribution function, in contrast...

In this paper, we provide an elaboration on the desirable properties of statistical depths for functional data. Although a formal definition has been put forward in the literature, there are still several unclarities to be tackled, and further insights to be gained. Herein, a few interesting connections between the wanted properties are found. In p...

A major drawback of many established depth functionals is their ineffectiveness in identifying functions outlying merely in shape. Herein, a simple modification of functional depth is proposed to provide a remedy for this difficulty. The modification is versatile, widely applicable, and introduced without imposing any assumptions on the data, such...

A strong law of large numbers for continuous random functions, and associated tensor product surfaces is established in the setup of discretely observed functional data. The result is shown in the framework of uniform convergence of functions, and stated without imposing any distributional assumptions. It is demonstrated that, under mild conditions...

We consider copula modeling of the dependence between two or more random variables in the presence of a multivariate covariate. The dependence parameter of the conditional copula possibly depends on the value of the covariate vector. In this paper we develop a new testing methodology for some important parametric specifications of this dependence p...

The interest is in regression quantiles in varying coefficient models for analysing longitudinal data. The coefficients are allowed to vary with time, and the error variance (the variability function) varies with the covariates to allow for heteroscedasticity. The functional coefficients are estimated using penalized splines (P-splines), not requir...

We consider varying coefficient models which are an extension of the classical linear regression models in the sense that the regression coefficients are replaced by functions in certain variables (often time). Varying coefficient models have been popular in longitudinal data and panel data studies, and have been applied in fields, such as finance...

Varying coefficient models are flexible models to describe the dynamic structure in longitudinal data. Quantile regression, more than mean regression, gives partial information on the conditional distribution of the response given the covariates. In the literature, the focus has been so far mostly on homoscedastic quantile regression models, wherea...

In dependence modelling using conditional copulas, one often imposes the working assumption that the covariate influences the conditional copula solely through the marginal distributions. This so-called (pairwise) simplifying assumption is almost standardly made in vine copula constructions. However, in recent literature evidence was provided that...

Quantile regression is an important tool for describing the characteristics of conditional distributions. Population conditional quantile functions cannot cross for different quantile orders. Unfortunately estimated regression quantile curves often violate this and cross each other, which can be very annoying for interpretations and further analysi...

Quantile regression in varying-coefficient models (VCM) using one particular nonparametric technique called P-splines. The functions can be applied on three types of VCM; (1) Homoscedastic VCM, (2) Simple heteroscedastic VCM, and (3) General heteroscedastic VCM.

A general family of asymmetric distributions (a special case of two-pice distributions), in which the location parameter is a specific quantile of the distribution (which is the main attraction of this family) is introduced. We present the resulting special cases of asymmetric normal distributions, asymmetric students-t distributions and asymmetric...

The smoothness of Tukey depth contours is a regularity condition often encountered in asymptotic theory, among others. This condition ensures that the Tukey depth fully characterizes the underlying multivariate probability distribution. In this paper we demonstrate that this regularity condition is rarely satisfied. It is shown that even well-behav...

Several depths suitable for infinite-dimensional functional data that are available in the literature are of the form of an integral of a finite-dimensional depth function. These functionals are characterized by projecting functions into low-dimensional spaces, taking finite-dimensional depths of the projected quantities, and finally integrating th...

When describing adequately complex data structures one is often confronted with the fact that mean as well as variance (or more generally dispersion) is highly influenced by some covariates. Drawbacks of the available methods is that they are often based on approximations and hence a theoretical study should deal with also studying these approximat...

This paper concerns a robust variable selection method in multiple linear regression: the robust S-nonnegative garrote variable selection method. In this paper the consistency of the method, both in terms of estimation and in terms of variable selection, is established. Moreover, the robustness properties of the method are further investigated by p...

We consider bias reduced estimators for the tail index and tail probabilities under random right censoring in case of Pareto-type distributions. The solution is based on second-order refined peaks-over-threshold modelling as developed in Beirlant et al. (2009).

For a pair (Y1, Y2) of random variables there exist several measures of association that characterize the dependence between Y1 and Y2 by means of one single value. Classical examples are Pearson’s correlation coefficient, Kendall’s tau and Spearman’s rho. For the situation where next to the pair (Y1, Y2) there is also a third variable X present, s...

A new methodology for selecting a Bayesian network for continuous data outside the widely used class of multivariate normal distributions is developed. The ‘copula DAGs’ combine directed acyclic graphs and their associated probability models with copula C/D-vines. Bivariate copula densities introduce flexibility in the joint distributions of pairs...

A general result on weak convergence of the empirical measure of discretely observed functional data is shown. It is applied to the problem of estimation of functional mean value, and the problem of consistency of various types of depth for functional data. Counterexamples illustrating the fact that the assumptions as stated cannot be dropped easil...

Robust selection of variables in a linear regression model is investigated. Many variable selection methods are available, but very few methods are designed to avoid sensitivity to vertical outliers as well as to leverage points. The nonnegative garrote method is a powerful variable selection method, developed originally for linear regression but r...

In the analysis of functional data, the concept of data depth is of importance. Strong consistency of a sample version of a data depth is among the basic statistical properties that need to hold. In this paper we discuss consistency properties of three popular types of functional depth: the band depth, the half-region depth and the infimal depth. T...

This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y1 and Y2 given X, does not depend on the value of X. This latter indepen...

This paper concerns depth functions suitable for smooth functional data. We suggest a modification of the integrated data depth that takes into account the shape properties of the functions. This is achieved by including a derivative(s) into the definition of the suggested depth measures. We then further investigate the use of integrated data depth...

Selecting among a large set of variables those that influence most a response variable is an important problem in statistics.When the assumed regression model involves a nonparametric component, penalized regression techniques, and in particular P-splines, are among the commonly used methods. The aim of this paper is to provide a brief review of va...

In this chapter we first review recent developments in the use of copulas for studying dependence structures between variables. We discuss and illustrate the concepts of unconditional and conditional copulas and association measures, in a bivariate setting. Statistical inference for conditional and unconditional copulas is discussed, in various mod...

We investigate distributional properties of the sum of dd possibly unbounded random variables. The joint distribution of the random vector is formulated by means of an absolutely continuous copula, allowing for a variety of different dependence structures between the summands. The obtained expression for the distribution of the sum features a separ...

Additive varying coefficient models are a natural extension of multiple linear regression models, allowing the regression coefficients to be functions of other variables. Therefore these models are more flexible to model more complex dependencies in data structures. In this paper we consider the problem of selecting in an automatic way the signific...

Quantile regression, as a generalization of median regression, has been widely used in statistical modeling. To allow for analyzing complex data situations, several flexible regression models have been introduced. Among these are the varying coefficient models, that differ from a classical linear regression model by the fact that the regression coe...

The paper deals with non-parametric estimation of a conditional distribution function. We suggest a method of preadjusting the original observations non-parametrically through location and scale, to reduce the bias of the estimator.We derive the asymptotic properties of the estimator proposed. A simulation study investigating the finite sample perf...

Many univariate robust estimators are based on quantiles. As already theoretically pointed out by Fernholz (1997), smoothing the empirical distribution function with an appropriate kernel and bandwidth can reduce the variance and mean squared error (MSE) of some quantile-based estimators in small data sets. In this
paper we apply this idea on sever...

In this paper the interest is in testing for tail monotonicity dependence structures between two random variables. The main focus in the presentation of the statistical methodology is on left tail decreasingness, but the developed procedures can also be used for testing for other specific tail monotonicity dependence structures. In order to assess...

Positive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution o...

Testing homogeneity of dispersions may be of its own scientific interest as well as an important auxiliary step verifying assumptions of a main analysis. The problem is that many biological and ecological data are highly skewed and zero-inflated. Also the number of variables often exceeds the sample size. Thus data analysts often do not rely on par...

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional associa...

We present a fully automated framework to estimate derivatives nonparametrically without estimating the regression function. Derivative estimation plays an important role in the exploration of structures in curves (jump detection and discontinuities), comparison of regression curves, analysis of human growth data, etc. Hence, the study of estimatin...

The objective of this paper is to introduce the break-preserving local linear (BPLL) estimator for the estimation of unstable volatility functions for independent and asymptotically independent processes. Breaks in the structure of the conditional mean and/or the volatility functions are common in Finance. Nonparametric estimators are well suited f...

This article extends the nonnegative garrote method to a component selection method in a nonparametric additive model in which each univariate function is estimated with P-splines. We also establish the consistency of the procedure. An advantage of P-splines is that the fitted function is represented in a rather small basis of B-splines. A numerica...

In this paper the interest is to estimate the dependence between two variables conditionally upon a covariate, through copula modelling. In recent literature nonparametric estimators for conditional copula functions in case of a univariate covariate have been proposed. The aim of this paper is to nonparametrically estimate a conditional copula when...

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 lea...

In this article, we consider nonparametric smoothing and variable selection in varying-coefficient models. Varying-coefficient models are commonly used for analyzing the time-dependent effects of covariates on responses measured repeatedly (such as longitudinal data). We present the P-spline estimator in this context and show its estimation consist...

In Desmet and Gijbels (20094.
Desmet , L. , Gijbels , I. ( 2009 ). Local linear fitting and improved estimation near peaks . Canad. J. Statist. 37 ( 3 ): 453 – 475 . [CrossRef], [Web of Science ®]View all references), the problem of curve fitting on functions with peaks was addressed and a method was proposed that was building further on the one us...

We analyse data on abortion rate (AR) in Italy with a particular focus on different behaviours in different regions in Italy. The aim is to try to reveal the relationship between the AR and several covariates that describe in some way the modernity of the region and the condition of the women there. The data are mostly underdispersed and the degree...

In the econometric literature on the estimation of production technologies, there has been considerable interest in estimating
so called cost frontier models that relate closely to models for extreme non-standard conditional quantiles (Aragon et al.
Econ Theor 21:358–389, 2005) and expected minimum input functions (Cazals et al. J Econometrics 106:...

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two n...

Generalized linear models are a widely used method to obtain parametric estimates for the mean function. They have been further extended to allow the relationship between the mean function and the covariates to be more flexible via generalized additive models. However, the fixed variance structure can in many cases be too restrictive. The extended...

One way to model a dependence structure is through the copula function which is a mean to capture the dependence structure in the joint distribution of variables. Association measures such as Kendall’s tau or Spearman’s rho can be expressed as functionals of the copula. The dependence structure between two variables can be highly influenced by a co...

A major aim in recent nonparametric frontier modeling is to estimate a partial frontier well inside the sample of production units but near the optimal boundary. Two concepts of partial boundaries of the production set have been proposed: an expected maximum output frontier of order m=1,2,... and a conditional quantile-type frontier of order [alpha...

The Kaplan-Meier estimator estimates the distribution function of a lifetime T based on a sample of randomly right censored observations. In survival analysis the lifetime T is a nonnegative random variable describing the time until a certain event of interest happens. In medical applications examples of such events are the time till death of a pat...

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to...

We study joint nonparametric estimators of the mean and the dispersion functions in extended double exponential family models.
The starting point is the exponential family and the generalized linear models setting. The extended models allow for both
overdispersion and underdispersion, or even a combination of both. We simultaneously estimate the di...

P-splines regression provides a flexible smoothing tool. In this paper we consider difference type penalties in a context
of nonparametric generalized linear models, and investigate the impact of the order of the differencing operator. Minimizing
Akaike’s information criterion we search for a possible best data-driven value of the differencing orde...

This paper proposes a robust forecasting method for non-stationary time series. The time series is modelled using non-parametric heteroscedastic regression, and fitted by a localized MM-estimator, combining high robustness and large efficiency. The proposed method is shown to produce reliable forecasts in the presence of outliers, non-linearity, an...

Linear least squares regression is among the most well known classical methods. This and other parametric least squares regression models do not perform well when the modeling is too restrictive to capture the nonlinear effect the covariates have on the response. Locally weighted least squares regression (loess) is a modern technique that combines...

P-splines regression is a flexible smoothing tool in which the starting point is a highly parameterised model and overfitting is prevented by introducing a penalty function. A common form of the penalty term is obtained by taking a prespecified order of differences of adjacent coefficients. This paper deals with a data-driven choice of the differen...

The analysis of censored data is a major issue in survival studies. Censored data are together with truncated data, missing data, current status data, and others, among the complex data structures in which only partial information on the variable(s) of interest is available. In this article, we discuss the various types of censoring mechanisms and...

In this chapter, we review the most important robust and/or nonparametric statistical methods. The focus is on methods that are used in comprehensive chemometrics. We review robust estimators of location and scale, and show how to adjust boxplots when dealing with skewed distributions and outlyingness. Robust estimators for correlation and covarian...

Generalized Linear Models are a widely used method to obtain parametric estimates for the mean function. They have been further extended to allow the relationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Q...

This paper deals with nonparametric estimation of an unknown density function which possibly is discontinuous or non-differentiable in an unknown finite number of points. Estimation of such irregular densities is accomplished by viewing the problem as a regression problem and applying recent techniques for estimation of irregular regression curves....

This paper combines recent developments in methods for solving and estimating rational expectations dynamic models. These developments are applied to a model of labor-market search, where firms operate under uncertainty. We assess the ability of the structural model to mimic nonlinear features found in the data. The solution to the model is obtaine...

We consider a nonparametric noisy data model Y k =f(x k )+ε k , k=1,⋯,n, where the unknown signal f:[0,1]→ℝ is assumed to belong to a wide range of function classes, including discontinuous functions, and the ε k ' s are independent, identically distributed noises with zero median. The distribution of the noise is assumed to be unknown and to satis...

The authors look into the problem of estimating regression functions that exhibit jump irregularities in the first derivative. They investigate the behaviour of the bias in the local linear fit and show the superior performance of appropriate one-sided versions of the local linear fit near such irregularities. They then propose an improved estimati...

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, Radulovi\v{c} 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...

It is a well-known problem that obtaining a correct bandwidth/smoothing parameter in nonparametric regression is difficult in the presence of correlated errors. There exist a wide variety of methods coping with this problem, but they all critically depend on a tuning procedure which requires accurate information about the correlation structure. We...

One of the popular method for fitting a regression function is regularization: minimizing an objective function which enforces
a roughness penalty in addition to coherence with the data. This is the case when formulating penalized likelihood regression
for exponential families. Most of the smoothing methods employ quadratic penalties, leading to li...

We propose two density estimators of the survival distribution in the setting of the Koziol-Green random-censoring model. The estimators are obtained by maximum-penalized-likelihood methods, and we provide an algorithm for their numerical evaluation. We establish the strong consistency of the estimators in the Hellinger metric, the Lp-norms, p= 1,2...

For nonparametric estimation of a smooth regression function, local linear fitting is a widely-used method. The goal of this
paper is to briefly review how to use this method when the unknown curve possibly has some irregularities, such as jumps or
peaks, at unknown locations. It is then explained how the same basic method can be used when estimati...

The behavior of the presmoothed density estimator is studied when different ways to estimate the conditional probability of
uncensoring are used. The Nadaraya–Watson, local linear and local logistic approach are compared via simulations with the
classical Kaplan–Meier estimator. While the local logistic presmoothing estimator presents the best perf...

Many statistical procedures involve calculation of integrals or optimization (minimization or maximization) of some objective
function. In practical implementation of these, the user often has to face specific problems such as seemingly numerical instability
of the integral calculation, choices of grid points, appearance of several local minima or...

In this paper we focus on nonparametric estimation of a constrained regression function using penalized wavelet regression techniques. This results into a convex optimization problem under linear constraints. Necessary and sufficient conditions for existence of a unique solution are discussed. The estimator is easily obtained via the dual formulati...

A class of estimating functions is introduced for the regression parameter of the Cox proportional hazards model to allow unknown failure statuses on some study subjects. The consistency and asymptotic normality of the resulting estimators are established under mild conditions. An adaptive estimator which achieves the minimum variance-covariance bo...

This paper deals with nonparametric estimation of a regression curve, where the estimation method should preserve possible
jumps in the curve. At each point x at which one wants to estimate the regression function, the method chooses in an adaptive way among three estimates: a local
linear estimate using only datapoints to the left of x, a local li...

In this paper, we are interested in the problem of estimating a discontinuous surface from noisy data. A novel procedure for this problem is proposed based on local linear kernel smoothing, in which local neighborhoods are adapted to the local smoothness of the surface measured by the observed data. The procedure can therefore remove noise correctl...

We consider estimation of the boundary of the support of a density func-tion when only a contaminated sample from the density is available. This estimation problem is a necessary step when estimating a density with unknown support, dif-ferent from the whole real line, since then modifications of the usual kernel type estimators are needed for consi...

Estimation of the support of a density function is considered, when only a contaminated sample from the density is available. A kernel-based method has been proposed in the literature, where the authors study theoretical bias and variance of the estimator. Practical implementation issues of this method are considered here, which are a necessary sup...

We propose a sound approach to bandwidth selection in nonparametric kernel testing. The main idea is to find an Edgeworth expansion of the asymptotic distribution of the test concerned. Due to the involvement of a kernel bandwidth in the leading term of the Edgeworth expansion, we are able to establish closed-form expressions to explicitly represen...

The presmoothed density estimator makes use of presmoothing ideas replacing the in-dicators of no censoring by some preliminary nonparametric estimator of the conditional probability of uncensoring p (t). This conditional probability is in fact a conditional mean, and therefore a regression function. In this work the behavior of this presmoothed de...