Jens Wagener

Bridge Estimators and the Adaptive Lasso under Heteroscedasticity

In this paper we investigate penalized least squares methods in linear regression models with heteroscedastic error structure. It is demonstrated that the basic properties with respect to model selection and parameter estimation of bridge estimators, Lasso and adaptive Lasso do not change if the assumption of homoscedasticity is violated. However, these estimators do not have oracle properties in the sense of Fan and Li (2001) if the oracle is based on weighted least squares. In order to address this problem we introduce weighted penalized least squares methods and demonstrate their advantages by asymptotic theory and by means of a simulation study.

The adaptive lasso in high-dimensional sparse heteroscedastic models

In this paper we study the asymptotic properties of the adaptive Lasso estimate in high-dimensional sparse linear regression models with heteroscedastic errors. It is demonstrated that model selection properties and asymptotic normality of the selected parameters remain valid but with a suboptimal asymptotic variance. A weighted adaptive Lasso estimate is introduced and investigated. In particular, it is shown that the new estimate performs consistent model selection and that linear combinations of the estimates corresponding to the non-vanishing components are asymptotically normally distributed with a smaller variance than those obtained by the “classical” adaptive Lasso. The results are illustrated in a data example and by means of a small simulation study.

Censored quantile regression processes under dependence and penalization

We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalized estimators such as the adaptive lasso yield sub-optimal rates if the coecients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specic problems of censored data and yield estimates with an optimal rate. In contrast to most of the literature, the asymptotic analysis does not require the assumption of independent observations, but is based on rather weak assumptions, which are satised for many

The quantile process under random censoring

In this paper we discuss the asymptotic properties of quantile processes under random censoring. In contrast to most work in this area we prove weak convergence of an appropriately standardized quantile process under the assumption that the quantile regression model is only linear in the region, where the process is investigated. Additionally, we also discuss properties of the quantile process in sparse regression models including quantile processes obtained from the Lasso and adaptive Lasso. The results are derived by a combination of modern empirical process theory, classical martingale methods and a recent result of Kato (2009).

Least Squares Estimation in High Dimensional Sparse Heteroscedastic Models

This contribution gives a brief review on penalized least squares methods in sparse linear regression models with a specific focus on heteroscedastic data structures. We discuss the well known bridge estimators, Lasso and adaptive Lasso and a new class of weighted penalized least squares methods, which address the problem of heteroscedasticity. We give a careful explanation on how the choice of the regularizing parameter affects the quality of the statistical inference (such as conservative or consistent model selection). The new estimators are asymptotically (pointwise) as efficient as estimators which are assisted by a model selection oracle. The results are illustrated by means of a small simulation study and the analysis of a data example.

Nonparametric comparison of quantile curves: a stochastic process approach

A new test for comparing conditional quantile curves is proposed which is able to detect Pitman alternatives converging to the null hypothesis at the optimal rate. The basic idea of the test is to measure differences between the curves by a process of integrated nonparametric estimates of the quantile curve. We prove weak convergence of this process to a Gaussian process and study the finite sample properties of a Kolmogorov–Smirnov test by means of a simulation study.

Nonparametric analysis of covariance using quantile curves

We consider the problem of testing the equality of J quantile curves from independent samples. A test statistic based on an L2-distance between non-crossing nonparametric estimates of the quantile curves from the individual samples is proposed. Asymptotic normality of this statistic is established under the null hypothesis, local and fixed alternatives, and the finite sample properties of a bootstrap based version of this test statistic are investigated by means of a simulation study. AMS Subject Classification: 62G10, 62G35