Semiparametric estimation of conditional heteroscedasticitity via single-index modeling

Abstract

We consider a single-index structure to study heteroscedasticity in regression with high-dimensional predictors. A general class of estimating equations is introduced, the resulting estimators remain consistent even when the structure of the variance function is misspecified. The proposed estimators also possess an adaptive property in an asymptotic sense. That is, they estimate the conditional variance function asymptotically as well as if the conditional mean function was given a priori. Numerical studies confirm our theoretical observations and demonstrate that our proposed estimator is superior to existing estimators with less bias and smaller standard deviation.

Conditional median absolute deviation

“…, n}. This echoes the phenomenon observed in many existing works, such as Fan and Yao, [11] Zhu et al, [16] etc. This phenomenon is termed as the adaptiveness property in Zhu et al [16] Yet, in the present context we demonstrate the adaptiveness property in the robust estimation context.…”

“…It is resilient to the presence of outliers and extreme variables in characterizing the local dispersion. This phenomenon makes our present work very distinctive from existing literature in terms of describing the local variability, such as [10][11][12][13][14][15][16] and references therein. This paper is organized as follows.…”

Efficient Estimation in Heteroscedastic Varying Coefficient Models

“…Keilegom and Wang [16] considered a general class of mean-variance regression models, in which both the mean function and the variance function were semiparametrically modeled. Zhu et al [17] consider a single-index structure to study heteroscedasticity in a single-index regression model with high-dimensional predictors.…”

Semiparametric Estimation and Inference of Variance Function with Large Dimensional Covariates

“…See, for example, Zhu (2009), Zhu, Dong andLi (2013) and Luo, Li and Yin (2014) for such two step estimation procedures. Obtaining residuals requires consistent estimation of the conditional mean or the central mean subspace, where many existing methods apply (Li and Duan 1989, Li 1992, Ichimura 1993, Cook and Li, 2002, Xia et al 2002, Ma and Zhu 2014, Luo, Li and Yin 2014.…”