Least-trimmed squares: asymptotic normality of robust estimator in semiparametric regression models

Abstract

In classical regression analysis, the ordinary least-squares estimation is the best method if the essential assumptions are met to obtain regression weights. However, if the data do not satisfy some of these assumptions, then results can be misleading. Especially, outliers violate the assumption of normally distributed residuals in the least-squares regression. So, it is important to use methods of estimation designed to tackle these problems. Robust regression is an important method for analysing data that are contaminated with outliers. In this paper, under some non-stochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized least-trimmed squares (LTS) estimators for the regression parameter is proposed. The LTS method is a highly robust regression estimator based on the subset of h observations (out of n). For practical use, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Asymptotic normality and n-consistency of proposed robust estimators under some conditions are also proved and a robust test is given for testing the symmetry hypothesis Ho:Rβ=0. Through the Monté-Carlo simulation studies and a real data example, performance of the feasible type of robust estimators are compared with the classical ones in restricted semiparametric regression models.