On regularization methods based on Rényi's pseudodistances for sparse high-dimensional linear regression models

Abstract

Several regularization methods have been considered over the last decade for sparse high-dimensional linear regression models, but the most common ones use the least square (quadratic) or likelihood loss and hence are not robust against data contamination. Some authors have overcome the problem of nonrobustness by considering suitable loss function based on divergence measures (e.g., density power divergence, γ−divergence, etc.) instead of the quadratic loss. In this paper we shall consider a loss function based on the Rényi's pseudodistance jointly with non-concave penalties in order to simultaneously perform variable selection and get robust estimators of the parameters in a high-dimensional linear regression model of nonpolynomial dimensionality. The desired oracle properties of our proposed method are derived theoretically and its usefulness is illustustrated numerically through simulations and real data examples.

Robust approach for comparing two dependent normal populations through Wald-type tests based on Rényi's pseudodistance estimators

“…Theorem 1 The expression of the variance-covariance matrix in the asymptotic distribution, (7), is given by (8) where…”

“…In the same vein, Castilla et al (2020a) introduced Wald-type tests based on the minimum RPD estimators for the MRM and its extension for Generalized Linear models was presented in Jaenada and Pardo (2021). Further, Castilla et al (2020b) studied the MRPDE for the linear regression model in the ultra-high dimensional set-up.…”

Robust approach for comparing two dependent normal populations through Wald-type tests based on Rényi’s pseudodistance estimators

“…In the same vein, Castilla et al (2020a) introduced Wald-type tests based on the minimum RPD estimators for the MRM and its extension for Generalized Linear models was presented in Pardo (2021, 2022). Further, Castilla et al (2020b) studied the MRPDE for the linear regression model in the ultra-high dimensional set-up.…”