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An important complexity in censored data is that only partial information on the variables of interest is observed. In recent years, a large family of asymmetric distributions and maximum likelihood estimation for the parameters in that family has been studied, in the complete data case. In this paper, we exploit the appealing family of quantile-based asymmetric distributions to obtain flexible distributions for modelling right censored survival data. The flexible distributions can be generated using a variety of symmetric distributions and monotonic link functions. The interesting feature of this family is that the location parameter coincides with an index-parameter quantile of the distribution. This family is also suitable to characterize different shapes of the hazard function (constant, increasing, decreasing, bathtub and upside-down bathtub or unimodal shapes). Statistical inference is done for the whole family of distributions. The parameter estimation is carried out by optimizing a non-differentiable likelihood function. The asymptotic properties of the estimators are established. The finite-sample performance of the proposed method and the impact of censorship are investigated via simulations. Finally, the methodology is illustrated on two real data examples (times to weaning in breast-fed data and German Breast Cancer data).

ispartof: Econometrics and Statistics status: accepted

Early Access: MAR 2021; National audience; Regression extremiles define a least squares analogue of regression quantiles. They are determined by weighted expectations rather than tail probabilities. Of special interest is their intuitive meaning in terms of expected minima and maxima. Their use appears naturally in risk management where, in contrast to quantiles, they fulfill the coherency axiom and take the severity of tail losses into account. In addition, they are comonotonically additive and belong to both the families of spectral risk measures and concave distortion risk measures. This paper provides the first detailed study exploring implications of the extremile terminology in a general setting of presence of covariates. We rely on local linear (least squares) check function minimization for estimating conditional extremiles and deriving the asymptotic normality of their estimators. We also extend extremile regression far into the tails of heavy-tailed distributions. Extrapolated estimators are constructed and their asymptotic theory is developed. Some applications to real data are provided.

When the interest is in studying conditional dependencies, and more precisely the strength of conditional dependencies, some kind of averaging over the conditioning random vector may be needed. Examples of average measures that can serve in this context are the average conditional Kendall’s tau and partial Kendall’s tau. It is known that these measures differ in general. Some statistical tests are based on these average measures, and a better knowledge of them is of importance. The aim of this paper is to provide a quantitative study of the possible differences of these two average measures, and to establish su˚cient conditions under which they coincide. Both measures are studied in two fairly general settings. In each setting theoretical results are established as well as several illustrative examples given.

A new way of constructing flexible and unimodal circular models, focusing on the modal direction, is proposed. Starting from a base symmetric density and a weight function, a two–piece four parameters density is introduced. The proposed density provides an extension of the base density to allow for sharply peaked and flat–topped unimodal distributions as well as a wide range of skewness. In particular, it generalizes some well–known peakedness–free models such as the Batschelet and Papakonstantinou densities. The four parameters of the model have a clear interpretation: modal direction, concentration, peakedness at the left and at the right of the modal direction. Symmetric submodels are obtained when the peakedness parameters are equal. The main properties related to the shape of the new density are presented and asymptotic results for maximum likelihood estimators are derived. An illustrative application concerning the flight orientation of migrating raptors is investigated.

This paper studies local polynomial estimation of expectile regression. Expectiles and quantiles both provide a full characterization of a (conditional) distribution function, but have each their own merits and inconveniences. Local polynomial fitting as a smoothing technique has a major advantage of being simple, allowing for explicit expressions and henceforth advantages when doing inference theory. The aim of this paper is twofold: to study in detail the use of local polynomial fitting in the context of expectile regression and to contribute to the important issue of bandwidth selection, from theoretical and practical points of view. We discuss local polynomial expectile regression estimators and establish an asymptotic normality result for them. The finite-sample performance of the estimators, combined with various bandwidth selectors, is investigated in a simulation study. Some illustrations with real data examples are given.

ispartof: ANNALS OF STATISTICS vol:50 issue:2 pages:754-780 status: published

This article studies M-type estimators for fitting robust additive models in the presence of anomalous data. The components in the additive model are allowed to have different degrees of smoothness. We introduce a new class of wavelet-based robust M-type estimators for performing simultaneous additive component estimation and variable selection in such inhomogeneous additive models. Each additive component is approximated by a truncated series expansion of wavelet bases, making it feasible to apply the method to nonequispaced data and sample sizes that are not necessarily a power of 2. Sparsity of the additive components together with sparsity of the wavelet coefficients within each component (group), results into a bi-level group variable selection problem. In this framework, we discuss robust estimation and variable selection. A two-stage computational algorithm, consisting of a fast accelerated proximal gradient algorithm of coordinate descend type, and thresholding, is proposed. When using nonconvex redescending loss functions, and appropriate nonconvex penalty functions at the group level, we establish optimal convergence rates of the estimates. We prove variable selection consistency under a weak compatibility condition for sparse additive models. The theoretical results are complemented with some simulations and real data analysis, as well as a comparison to other existing methods.

International audience; In functional data the interest is to find a global mean pattern, but also to capture the individual curve differences in phase and amplitude. This can be done conveniently by building in random effects on two levels: in the warping functions to account for individual phase variations; and in the linear structure to deal with individual amplitude variations. Via an appropriate choice of the warping function and B-spline approximations, estimation in the nonlinear mixed effects functional model is feasible, and does not require any prior knowledge on landmarks for the functional data. Sufficient and necessary conditions for identifiability of the flexible model are provided. A theoretical study is conducted: we establish asymptotic normality and consistency of the estimators of the registration and amplitude models, convergence of the iterative process, and consistency of the final estimator provided by the iterative process. The finite-sample performance of the proposed estimation procedure is investigated in a simulation study, which includes comparisons with existing methods. The added value of the developed method is further illustrated via the analysis of a real data example.

This chapter deals with partially linear expectile regression using local polynomial fitting as a basic smoothing technique in the various steps. The advantage of the estimation method is that an explicit expression for an optimal choice of the bandwidth (matrix) can be established, and based on this, a rule-of-thumb bandwidth selector is presented. A small simulation study demonstrates that the estimation method with this data-driven choice of the bandwidth performs very well. An illustration with a real data example is provided.

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 penalized estimation procedures are shown to enjoy theoretically the oracle property both in the classical finite dimensional case as well as the high-dimensional case when the number of variables p is not fixed but can grow with the sample size n, and to achieve optimal asymptotic rates of convergence. A fast accelerated proximal gradient algorithm, of coordinate descent type, is proposed and implemented for computing the estimates and appears to be surprisingly efficient in solving the corresponding regularization problems including the case for ultra high-dimensional data where $$p \gg n$$ . Finally, a very extensive simulation study and some real data analysis, compare several recent existing M-estimation procedures with the ones proposed in the paper, and demonstrate their utility and their advantages.

We investigate the inference of the copula parameter in the semiparametric Gaussian copula model when the copula component, subject to the influence of a covariate, is only indirectly observed as the response in a linear regression model. We consider estimators based on residual ranks instead of the usual but unobservable oracle ranks. We first study two such estimators for the copula correlation matrix, one via inversion of Spearman’s rho and the other via normal scores rank correlation estimator. We show that these estimators are asymptotically equivalent to their counterparts based on the oracle ranks. Then, for the copula correlation matrix under constrained parametrizations, we show that the classical one-step estimator in conjunction with the residual ranks remains semiparametrically efficient for estimating the copula parameter. The accuracy of the estimators based on residual ranks is confirmed by simulation studies.

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 estimation. In a similar fashion, a broad family of conditional densities is considered in the regression setting. We discuss estimation of the parameters in the unconditional case, and establish an asymptotic normality result, with explicit expression for the asymptotic variance–covariance matrix. In the regression setting, we allow for flexible modelling and estimate nonparametrically the location and scale functions, leading to semiparametric estimation of conditional quantiles, again in the unifying framework of the considered broad family. The practical use of the proposed methods is illustrated in a real data application on locomotor performance in small and large terrestrial mammals.

© 2018 Oxford University Press. All rights reserved. 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 design setting and addresses the problem in situations where no prior knowledge about the correlation structure is available.We establish the asymptotic optimality of our proposed bandwidth selection criterion based on kernels K satisfying K(0) = 0. ispartof: Biometrika vol:105 issue:3 pages:681-690 status: published

Conditional copulas describe the conditional dependence and the influence that covariates have on the dependence structure between two (or more) variables. Of interest is to test the null hypothesis that the covariates have a specific effect. This paper proposes an omnibus test for testing the null hypothesis of a specified effect of the covariates. The test statistic is designed for having power against many alternatives, and can be used to test for a variety of covariate effects (no effects, linear effects, partial effects, etc.). A special case is the testing problem that the covariates do not affect the dependence structure. In this semiparametric framework the marginal distribution functions are estimated using nonparametric kernel techniques and the parametric dependence model is estimated using maximum likelihood estimation. We establish the asymptotic distribution of the test statistic under the null hypothesis, and evaluate the finite-sample performance of the test via a simulation study, which also includes comparisons with alternative tests. A real data analysis illustrates the practical use of the test.

© 2017 The Korean Statistical Society 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, laws of large numbers for continuously observed functional data imply the corresponding laws under the discrete observational design of functions. Applications to the problem of estimation of expectation functions and covariance surfaces for discretely observed functional data are discussed. ispartof: Journal of the Korean Statistical Society vol:46 issue:4 pages:562-572 status: published

© 2017 Statistical Society of Canada 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 to the mean which is just one of its characteristic quantities. Regression quantiles substantially out-perform the least-squares estimator for a wide class of non-Gaussian error distributions. In this article we consider quantile varying coefficient models (VCMs) that are an extension of classical quantile linear regression models, in which one allows the coefficients to depend on other variables. We consider VCMs with various structures for the variance of the errors (the variability function) in order to allow for heteroscedasticity. For longitudinal data, the time (T) dependent coefficient functions in the signal and the variability functions are estimated with P-splines (Penalized B-splines). Consistency of the proposed estimators is proved. Further, likelihood-ratio-type tests are considered for comparing the variability functions. The performance of the testing procedure is illustrated on simulated and real data. The Canadian Journal of Statistics 46: 246–264; 2018 © 2017 Statistical Society of Canada. ispartof: The Canadian Journal of Statistics vol:46 issue:2 pages:246-264 status: published

Quantile regression is an important tool in data analysis. Linear regression, or more generally, parametric quantile regression imposes often too restrictive assumptions. Nonparametric regression avoids making distributional assumptions, but might have the disadvantage of not exploiting distributional modelling elements that might be brought in. A semiparametric approach towards estimating conditional quantile curves is proposed. It is based on a recently studied large family of asymmetric densities of which the location parameter is a quantile (and not a mean). Passing to conditional densities and exploiting local likelihood techniques in a multiparameter functional setting then leads to a semiparametric estimation procedure. For the local maximum likelihood estimators the asymptotic distributional properties are established, and it is discussed how to assess finite sample bias and variance. Due to the appealing semiparametric framework, one can discuss in detail the bandwidth selection issue, and provide several practical bandwidth selectors. The practical use of the semiparametric method is illustrated in the analysis of maximum winds speeds of hurricanes in the North Atlantic region, and of bone density data. A simulation study includes a comparison with nonparametric local linear quantile regression as well as an investigation of robustness against miss-specifying the parametric model part.

© 2017 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. 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 as differentiability. It is shown that many favorable attributes of the original depths for functions, including consistency properties, remain preserved for the modified depths. The powerfulness of the new approach is demonstrated on a number of examples for which the known depths fail to identify the outlying functions. Supplementary material for this article is available online. ispartof: Journal of Computational and Graphical Statistics vol:26 issue:4 pages:883-893 status: published

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 providing a lower bound for the breakdown point, and by deriving the influence function. The provided expressions nicely reveal the impact that the choice of an initial estimator has on the robustness properties of the variable selection method. Illustrative examples of influence functions for the S-nonnegative garrote as well as for the original (non-robust) nonnegative garrote variable selection method are provided. This research was supported by the IAP Research Network IAP P7/06 of the Belgian State (Belgian Science Policy), project GOA/12/014 of the KU Leuven Research Fund, and grant no. Q2 1.5.137.13N of the Flemish Research Foundation (FWO).

The interest is in regression quantiles in varying coefficient models for analyzing 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 requiring specification of the error distribution. A likelihood-ratio-type test is considered to test the shape (constancy, monotonicity and/or convexity) of the functional coefficients. Further, testing procedures based on L1-norm, L2-norm and L∞-norm of the differences of the P-splines coefficients are considered to test for constant functional coefficients. These norm based tests perform better than the likelihood-ratio-type test in our simulation study. An extreme value test for testing monotonicity or convexity, also performs better than the likelihood-ratio-type test. The likelihood-ratio-type test is, however, useful when testing the shape of the coeffi- cients in signal and in variability function simultaneously. A real data example demonstrates the testing procedures. The authors gratefully acknowledge support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). I. Gijbels acknowledges support from the KU Leuven Research Council (GOA/12/014). A. Verhasselt acknowledges support from the The authors gratefully acknowledge support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). I. Gijbels acknowledges support from the KU Leuven Research Council (GOA/12/014). A. Verhasselt acknowledges support from the Fonds Wetenschappelijk Onderzoek (FWO) research grant 1518917N. M.A. Ibrahim and A. Verhasselt acknowledge support from the Special Research Fund (BOF) of Hasselt University [14NO6VHAA]. The infrastructure of the VSC-Flemish Supercomputer Center, funded by the Hercules Foundation and the Flemish Government - department EWI, was used for the simulations.

© 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group. 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 heteroscedastic linear models. The paper gives essential contributions in the area of simultaneous robust estimation and variable selection, relying on basics of the nonnegative garrote method which has been proven to have very good practical as well as theoretical properties in the homoscedastic linear model context. Several numerical examples, simulations and analysis of real data, demonstrate the performances and practical use of the discussed methods. Moreover, we provide expressions for the influence functions of the estimators of the mean and the error variance parameters. Influence functions are plotted in a simple setting providing insights in the sensitivity of the estimators for a single outlying observation. ispartof: STATISTICS vol:53 issue:3 pages:489-532 status: published

© 2018, © 2018 American Statistical Association. 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 article considers a valuable alternative class to expectiles, called extremiles, which parallels the class of quantiles and includes the family of expected minima and expected maxima. The new class is motivated via several angles, which reveals its specific merits and strengths. Extremiles suggest better capability of fitting both location and spread in data points and provide an appropriate theory that better displays the interesting features of long-tailed distributions. We discuss their estimation in the range of the data and beyond the sample maximum. A number of motivating examples are given to illustrate the utility of estimated extremiles in modeling noncentral behavior. There is in particular an interesting connection with coherent measures of risk protection. Supplementary materials for this article are available online. ispartof: Journal of the American Statistical Association vol:114 issue:527 pages:1366-1381 status: published

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 provided. The results under the general framework are then applied to some specific examples of asymmetric densities. The use of the asymmetric densities is illustrated in a real‐data analysis. This research was supported by the FWO (Fonds Wetenschappelijk Onderzoek) research project G.0826.15N (Flemish Science Foundation) and the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy; Federaal Wetenschapsbeleid). The first author gratefully acknowledges support from the GOA/12/014 project of the Research Fund KU Leuven (Onderzoeksraad, KU Leuven). The third author acknowledges support from the Flemish Science Foundation (FWO research grant 1518917N) and from the Special Research Fund (Bijzonder Onderzoeksfonds) of Hasselt University.

In this paper the interest is to elaborate on the generalization of bivariate association measures, namely Spearman’s rho, Kendall’s tau, Blomqvist’s beta and Gini’s gamma, for a general dimension d ≥ 2 . Desirable properties and axioms for such generalizations are discussed, where special attention is given to the impact of the addition of: (i) an independent random variable to a random vector; (ii) a conical combination of all components; (iii) a set of arbitrary random components. Existing generalizations are evaluated with respect to the axiom set. For a d -variate Gini’s gamma, a simplified formula is developed, making its analytical computation easier. Further, for Archimedean and meta-elliptical copulas the asymptotic behaviour when the dimension d increases is studied. Nonparametric estimation of the considered generalizations of multivariate association measures is reviewed and a nonparametric estimator of the multivariate Gini’s gamma is introduced. The practical use of multivariate association measures is illustrated on a real data example.

© 2016 Taylor & Francis. 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-behaved probability distributions with symmetrical, smooth and (strictly) quasi-concave densities may have non-smooth Tukey depth contours, and that the smoothness behaviour of depth contours is fairly unpredictable. peerreview_statement: The publishing and review policy for this title is described in its Aims & Scope. aims_and_scope_url: http://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=gsta20 ispartof: Statistics vol:50 issue:5 pages:1075-1085 status: published

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).

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 thorough study of properties of existing multivariate tail coefficients in the light of a set of desirable properties, (ii) proposing some new multivariate tail measurements, (iii) dealing with estimation of the discussed coefficients and establishing asymptotic consistency, and, (iv) studying the behavior of tail measurements with increasing dimension of the random vector. A set of illustrative examples is given, and practical use of the tail measurements is demonstrated in a data analysis with a focus on dependencies between stocks that are part of the EURO STOXX 50 market index.

© 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group. We provide a computational framework for the selection of weights (W1, . . .,Wd) that minimize the expected shortfall of the aggregated risk Z= ∑i=id WiXi. 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 continuous random variable with finite expectation, the dependence structure can be modelled by any absolutely continuous copula function. A real-world application to portfolio selection illustrates the usability of the new framework. ispartof: Applied Mathematical Finance vol:25 issue:1 pages:66-106 status: published

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 analysis. In this paper we are concerned with flexible varying-coefficient modelling, and develop methods for quantile regression that ensure that the estimated quantile curves do not cross. A second aim of the paper is to allow for some heteroscedasticity in the error modelling, and to also estimate the associated variability function. We investigate the finite-sample performances of the discussed methods via simulation studies. Some applications to real data illustrate the use of the methods in practical settings. This research is supported by the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy), and project GOA/12/014 of the Research Fund of the KU Leuven. The authors thank the Editor, an Associate Editor and the anonymous reviewers for their very valuable comments which led to a considerable improvement of the paper.

© Institute of Mathematical Statistics, 2017. 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 particular, it is demonstrated that the conditions needed for some desirable properties to hold are extremely demanding, and virtually impossible to be met for common depths. We establish adaptations of these properties which prove to be still sensible, and more easily met by common functional depths. ispartof: Statistical Science vol:32 issue:4 pages:630-639 status: published

© 2015 Elsevier Inc. 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. The latter is a special case of the recently introduced general class of Φ-depths. All three considered depth functions are of a non-integrated type. Counterexamples illustrate some problems with consistency results for these data depths. The main contribution of this paper consists of providing sufficient conditions for consistency of these non-integrated data depths to hold. publisher: Elsevier articletitle: Consistency of non-integrated depths for functional data journaltitle: Journal of Multivariate Analysis articlelink: http://dx.doi.org/10.1016/j.jmva.2015.05.012 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved. ispartof: Journal of Multivariate Analysis vol:140 pages:259-282 status: published

© 2014 Elsevier B.V. All rights reserved. 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 recently successfully extended to more complex regression models. The method has good performances and its theoretical properties have been established. The aim is to robustify the nonnegative garrote method for linear regression as to make it robust to vertical outliers and leverage points. Several approaches are discussed, and recommendations towards a final good performing robust nonnegative garrote method are given. The proposed method is evaluated via a simulation study that also includes a comparison with existing methods. The method performs very well, and often outperforms existing methods. A real data application illustrates the use of the method in practice. publisher: Elsevier articletitle: Robust nonnegative garrote variable selection in linear regression journaltitle: Computational Statistics & Data Analysis articlelink: http://dx.doi.org/10.1016/j.csda.2014.11.009 content_type: article copyright: Copyright © 2014 Elsevier B.V. All rights reserved. ispartof: Computational Statistics & Data Analysis vol:85 pages:1-22 status: published

© 2015, Springer-Verlag Berlin Heidelberg. 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 depths in supervised classification problems. The performances of classification rules based on different data depths are investigated, both in simulated and real data sets. As the proposed depth function provides a natural alternative to the depth function based on random projections, the difference in the performances of these two methods are discussed in more detail. ispartof: Computational Statistics vol:30 issue:4 pages:1011-1031 status: published

We investigate distributional properties of the sum of d 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 separation property into marginal and dependence structure contributions typical for copula approaches. Along the same lines we obtain the formulation of a conditional expectation closely related to the expected shortfall common in actuarial and financial literature. We further exploit the separation to introduce new numerical algorithms to compute the distribution and quantile function, as well as this conditional expectation. A comparison with the most common competitors shows that the discussed Path Integration algorithm is the most suitable method for computing these quantities. In our example, we apply the theory to compute Value-at-Risk forecasts for a trivariate portfolio of index returns.

© 2016, Springer-Verlag Berlin Heidelberg. 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, whereas there is an interest in looking into heteroscedastic modelling. This paper contributes to the area by modelling the heteroscedastic structure and estimating it from the data, together with estimating the quantile functions. The use of the proposed methods is illustrated on real-data applications. The finite-sample behaviour of the methods is investigated via a simulation study, which includes a comparison with an existing method. ispartof: Advances in Statistical Analysis vol:101 issue:2 pages:151-176 status: published

© 2016 Informa UK Limited, trading as Taylor & Francis Group. 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 such an assumption might not be justified. Among the first issues is thus to test for its appropriateness. In this paper nonparametric tests for the null hypothesis of the simplifying assumption are proposed, and their asymptotic behaviours, under the null hypothesis and under some local alternatives, are established. The tests are fully nonparametric in nature: not requiring choices of copula families nor knowledge of the marginals. In a simulation study, the finite-sample size and power performances of the tests are investigated, and compared with these of the few available tests. A real data application illustrates the use of the tests. peerreview_statement: The publishing and review policy for this title is described in its Aims & Scope. aims_and_scope_url: http://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=gsta20 ispartof: Statistics vol:51 issue:3 pages:475-509 status: published

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 and health sciences. We estimate the coefficient functions by splines. An important question in a varying coefficient model is whether a coefficient function is monotone or convex. We develop consistent testing procedures for monotonicity and convexity. Moreover, we provide procedures to test simultaneously the shapes of certain coefficient functions in a varying coefficient model. The tests use constrained and unconstrained regression splines. The performances of the proposed tests are illustrated on simulated data. We also give a real data application. We would like to thank the Editor, an Associate Editor, and the referees for their detailed reading and very valuable comments on the manuscript. M. Ahkim's research was supported by the Special Research Fund (BOF) of Universiteit Antwerpen (Grant Number 42FA070300FFB5994). A. Verhasselt and I. Gijbels gratefully acknowledge support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). I. Gijbels acknowledges support from the KU Leuven Research Council (GOA/12/014). The infrastructure of the VSC-Flemish Supercomputer Center, funded by the Hercules Foundation and the Flemish Government-department EWI, was used for the simulations.

© 2017 Elsevier Inc. 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 parameter: constant, linear, quadratic, etc. in the covariate values, possibly after transformation with a link function. The margins are left unspecified. Our novel methodology opens plenty of new possibilities for testing how the conditional copula depends on the multivariate covariate and also for variable selection in copula model building. The suggested test is based on a Rao-type score statistic and regularity conditions are given under which the test has a limiting chi-square distribution under the null hypothesis. For small and moderate sample sizes, a permutation procedure is suggested to assess significance. In simulations it is shown that the test performs well (even under misspecification of the copula family and/or the dependence parameter structure) in comparison to available tests designed for testing for constancy of the dependence parameter. The test is illustrated on a real data set on concentrations of chemicals in water samples. ispartof: Journal of Multivariate Analysis vol:159 pages:111-133 status: published

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 variable selection methods using P-splines. Starting from multiple linear regression models, with least-squares regression, and Ridge regression, we review standard methods that perform variable selection, such as LASSO, nonnegative garrote, the SCAD method, etc. We briefly discuss a general framework of penalization and regularization methods. Going toward more flexible regression models, with some nonparametric components, we discuss P-splines estimation. For some examples of flexible regression models, we then review a few variable selection methods using P-splines. A brief discussion on grouped regularization techniques and on a robust variable selection method is given. Furthermore, we mention key ingredients in Bayesian approaches, and end the paper by drawing the attention to several other issues in variable selection with P-splines. Throughout the paper we provide some illustrations. WIREs Comput Stat 2015, 7:1-20. doi: 10.1002/wics.1327

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 coefficients are no longer constant but functions that vary with the value taken by another variable, such as for example, time. In this paper, we study quantile regression in varying coefficient models for longitudinal data. The quantile function is modeled as a function of the covariates and the main task is to estimate the unknown regression coefficient functions. We approximate each coefficient function by means of P-splines. Theoretical properties of the estimators, such as rate of convergence and an asymptotic distribution are established. The estimation methodology requests solving an optimization problem that also involves a smoothing parameter. For a special case the optimization problem can be transformed into a linear programming problem for which then a Frisch–Newton interior point method is used, leading to a computationally fast and efficient procedure. Several data-driven choices of the smoothing parameters are briefly discussed, and their performances are illustrated in a simulation study. Some real data analysis demonstrates the use of the developed method.

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 the p -values of the test statistic, we resample from a constrained copula estimator. This can be done in a nonparametric or in a parametric way. The main difficulty is the construction of a constrained estimator and the development of a resampling technique. The finite-sample performances of the proposed testing procedures are investigated in a simulation study and illustrations on real data examples are provided.

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 association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.

© 2016 EDP Sciences, SMAI. 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 these projected marginal depths over a preset collection of projections. In this paper, a general class of integrated depths for functions is considered. Several depths for functional data proposed in the literature during the last decades are members of this general class. A comprehensive study of its most important theoretical properties, including measurability and consistency, is given. It is shown that many, but not all, properties of the integrated depth are shared with the finite-dimensional depth that constitutes its building block. Some pending measurability issues connected with all integrated depth functionals are resolved, a broad new notion of symmetry for functional data is proposed, and difficulties with respect to consistency results are identified. A general universal consistency result for the sample depth version, and for the generalized median, for integrated depth for functions is derived. ispartof: ESAIM-Probability and Statistics vol:20 pages:95-130 status: published

© 2015 Elsevier Inc. 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 easily are given. publisher: Elsevier articletitle: Weak convergence of discretely observed functional data with applications journaltitle: Journal of Multivariate Analysis articlelink: http://dx.doi.org/10.1016/j.jmva.2015.06.006 content_type: article copyright: Copyright © 2015 Elsevier Inc. All rights reserved. ispartof: Journal of Multivariate Analysis vol:146 pages:46-62 status: published