NONCONCAVE PENALIZED M-ESTIMATION WITH A DIVERGING NUMBER OF PARAMETERS

Abstract

M-estimation is a widely used technique for robust statistical inference. In this paper, we investigate the asymptotic properties of a nonconcave penalized M-estimator in sparse, high-dimensional, linear regression models. Compared with classic M-estimation, the nonconcave penalized M-estimation method can perform parameter estimation and variable selection simultaneously. The proposed method is resistant to heavy-tailed errors or outliers in the response. We show that, un- der certain appropriate conditions, the nonconcave penalized M-estimator has the so-called Oracle Property; it is able to select variables consistently, and the esti- mators of nonzero coefficients have the same asymptotic distribution as they would if the zero coefficients were known in advance. We obtain consistency and asymp- totic normality of the estimators when the dimension pn of the predictors satisfies the conditions pn log n/n → 0 and p 2/n → 0, respectively, where n is the sample size. Based on the idea of sure independence screening (SIS) and rank correla- tion, a robust rank SIS (RSIS) is introduced to deal with ultra-high dimensional data. Simulation studies were carried out to assess the performance of the proposed method for finite-sample cases, and a dataset was analyzed for illustration.