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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.

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.