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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.

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.

© 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.

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.

© 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

© 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

© 2016 Institute of Mathematical Statistics. All rights reserved. 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 approximations. This however is often ignored, making the statistical inference incomplete. In the proposed framework of double generalized modelling based on proper dispersion models we avoid this drawback and as such are in a good position to use recent results on Bregman divergence for establishing theoretical results for the proposed estimators in fairly general settings. We also study variable selection when there is a large number of covariates, with this number possibly tending to infinity with the sample size. The proposed estimation and selection procedure is investigated via a simulation study, that includes also a comparative study with competitors. The use of the methods is illustrated via some real data applications. ispartof: Electronic Journal of Statistics vol:10 issue:1 pages:1630-1676 status: published

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, so-called partial association measures, such as a partial Pearson’s correlation coefficient and a partial Kendall’s tau, have been proposed in the 1940’s. Following criticism on e.g. partial Kendall’s tau, better alternatives to these original partial association measures appeared in the literature: the conditional association measures, e.g. conditional Kendall’s tau, and conditional Spearman’s rho. Both, unconditional and conditional association measures can be expressed in terms of copulas. Even in case the dependence structure between Y1 and Y2 is influenced by a third variable X, we still want to be able to summarize the level of dependence by one single number. In this paper we discuss two different ways to do so, leading to two relatively new concepts: the (new concept of) partial Kendall’s tau, and the average Kendall’s tau. We provide a unifying framework for the diversity of concepts: global (or unconditional) association measures, conditional association measures, and partial and average association measures. The main contribution is that we discuss estimation of the newly-defined concepts: the partial and average copulas and association measures, and establish theoretical results for the estimators. The various concepts of association measures are illustrated on a real data example. This research was supported by the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). The first author gratefully acknowledges support from the GOA/12/014 project of the Research Fund KU Leuven. The second author gratefully acknowledges support from the grant GACR 15-04774Y. The third author is an extraordinary professor at the North-West University, Potchefstroom, South Africa. The authors are grateful to an Associate Editor and two referees for their valuable comments that led to a considerable improvement of the paper.

© 2015 Board of the Foundation of the Scandinavian Journal of Statistics. 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 independence often appears as a simplifying assumption in pair-copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location-scale models for the effect of the covariate X on the marginals of Y1 and Y2 and show that in these cases, weak convergence of the estimator, at n-rate, holds. The theoretical results are illustrated by simulations and a real data example. ispartof: Scandinavian Journal of Statistics vol:42 issue:4 pages:1109-1126 status: published

International audience; 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 significant variables among a large set of variables, when the interest is on a given response variable. In recent years several grouped regularization methods have been proposed and in this paper we present these under one unified framework in this varying coefficient model context. For each of the discussed grouped regularization methods we investigate the optimization problem to be solved, possible algorithms for doing so, and the variable and estimation consistency of the methods. We investigate the finite-sample performance of these methods, in a comparative study, and illustrate them on real data examples.

International audience; 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 numerical study illustrates the finite-sample performance of the method and includes a comparison with other methods. The nonnegative garrote method with P-splines has the advantage of being computationally fast and performs, with an appropriate parameter selection procedure implemented, overall very well. Real data analysis leads to interesting findings. Supplementary materials for this article (technical proofs, additional numerical results, R code) are available online.

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 for these events due to the flexibility of their functional form and their good asymptotic properties. However, the local polynomial kernel estimators are not consistent at points where the volatility function has a break. The estimator presented in this paper generalises the classical local linear (LL). The BPLL estimator maintains the desirable properties of the LL estimator with regard to the bias and the boundary estimation while it estimates the breaks consistently. An extensive Monte Carlo study is shown as well as detailed proofs of the estimator asymptotic behaviour.

International audience; 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 linear estimates, and are in general incapable of recovering discontinuities or other important attributes in the regression function. In contrast, non-linear estimates are generally more accurate. In this paper, we focus on non-parametric penalized likelihood regression methods using splines and a variety of non-quadratic penalties, pointing out common basic principles. We present an asymptotic analysis of convergence rates that justifies the approach. We report on a simulation study including comparisons between our method and some existing ones. We illustrate our approach with an application to Poisson non-parametric regression modeling of frequency counts of reported acquired immune deficiency syndrome (AIDS) cases in the UK.

In this paper the interest is in regression analysis for data that show possibly overdispersion or underdispersion. The starting point for modeling are generalized linear models in which we no longer admit a linear form for the mean regression function, but allow it to be any smooth function of the covariate(s). In view of analyzing overdispersed or underdispersed data, we additionally bring in an unknown dispersion function. The mean regression function and the dispersion function are then estimated using P-splines with difference type of penalty to prevent from overfitting. We discuss two approaches: one based on an extended quasi-likelihood idea and one based on a pseudo-likelihood approach. The choices of smoothing parameters and implementation issues are discussed. The performance of the estimation method is investigated via simulations and its use is illustrated on several data, including continuous data, counts and proportions. ispartof: Test vol:19 issue:3 pages:580-608 status: published

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. Moreover, the method can deal with estimation of densities that have an irregularity at the endpoint(s) of their support. A simulation study compares the performance of the proposed method with those of other methods available in the literature. A further illustration on real data is provided.

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 from a corner 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, Radulovi\v{c} 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., Comment: Published in at http://dx.doi.org/10.1214/08-AOS666 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Special issue: Multiscale Methods and Statistics: A Productive Marriage; International audience; We consider a nonparametric noisy data model $Y_k=f(x_k)+\epsilon_k$, $k=1, \ldots, n$, where the unknown signal $f: \ [0,1] \rightarrow{\mathbb{R}}$ is assumed to belong to a wide range of function classes, including discontinuous functions, and the $\epsilon_k's$ are independent identically distributed noises with zero median. The distribution of the noise is assumed to be unknown and to satisfy some weak conditions. Possible noise distributions may have heavy tails, so that, for example, no moments of the noise exist. The design points are assumed to be deterministic points, not necessarily equispaced within the interval $[0, 1]$. Since the functions can be nonsmooth and the noise may have heavy tails, traditional estimation methods (for example, kernel methods) cannot be applied directly in this situation. As in Brown, Cai, and Zhou (2008), our approach first uses local medians to construct certain variables $Z_k$ structured as a Gaussian nonparametric regression but, unlike in this paper, the resulting data being not equispaced, we apply a wavelet block penalizing procedure adapted to non-equidistant designs to construct an estimator of the regression function. Under mild assumptions on the design it is shown that our estimator simultaneously attains the optimal rate of convergence over a wide range of Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The performance of our procedures is evaluated on simulated data sets covering a broad variety of settings and on some data examples, and are compared with other proposals made in the literature for treating similar problems.

In this paper we consider a regression model with errors that are martingale differences. This modeling includes the regression of both independent and time series data. The aim is to study the appearance of structural breaks in both the mean and the variance functions, assuming that such breaks may occur simultaneously in both the functions. We develop nonparametric testing procedures that simultaneously test for structural breaks in the conditional mean and the conditional variance. The asymptotic distribution of an adaptive test statistic is established, as well as its asymptotic consistency and efficiency. Simulations illustrate the performance of the adaptive testing procedure. An application to the analysis of financial time series also demonstrates the usefulness of the proposed adaptive test in practice.

Summary Local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique. It extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces. Associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment. This paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation. The approach is then applied to least squares nonparametric regression and to nonparametric logistic regression. Our experiences in these two settings show that the general idea outlined here is powerful and encouraging.

This paper deals with censored data estimation of a general class of von Mises-type functionals of the survival time distribution $F$. Conditions are given under which an almost sure asymptotic representation holds for the estimator, obtained by applying the same functional to $\hat{F}_n$, the product-limit estimator of Kaplan and Meier.