Local Walsh-average-based estimation and variable selection for single-index models

• 来源期刊: 中国科学：数学（英文版）
• 年，卷（期）: 2019,（10）
• 页码范围: 1977-1996
• 页数: 20页
• 语种: 英文

摘要

We propose a robust estimation procedure based on local Walsh-average regression (LWR) for single-index models.Our novel method provides a root-n consistent estimate of the single-index parameter under some mild regularity conditions;the estimate of the unknown link function converges at the usual rate for the nonparametric estimation of a univariate covariate.We theoretically demonstrate that the new estimators show significant efficiency gain across a wide spectrum of non-normal error distributions and have almost no loss of efficiency for the normal error.Even in the worst case,the asymptotic relative efficiency (ARE) has a lower bound compared with the least squares (LS) estimates;the lower bounds of the AREs are 0.864 and 0.8896 for the single-index parameter and nonparametric function,respectively.Moreover,the ARE of the proposed LWR-based approach versus the ARE of the LS-based method has an expression that is closely related to the ARE of the signed-rank Wilcoxon test as compared with the t-test.In addition,to obtain a sparse estimate of the single-index parameter,we develop a variable selection procedure by combining the estimation method with smoothly clipped absolute deviation penalty;this procedure is shown to possess the oracle property.We also propose a Bayes information criterion (BIC)-type criterion for selecting the tuning parameter and further prove its ability to consistently identify the true model.We conduct some Monte Carlo simulations and a real data analysis to illustrate the finite sample performance of the proposed methods.