Semiparametric estimation for linear regression with symmetric errors

Abstract

To avoid the effect of distributional misspecification in the model-based regression, we propose an essentially nonparametric symmetric error distribution and construct a so-called doubly smoothed (DS) likelihood function by applying the same amount of smoothing to both the model and given data. To compute the DS maximum likelihood estimator based on the DS likelihood, we propose an approximated DS likelihood which has the form of a semiparametric mixture likelihood and apply some existing algorithms in the nonparametric mixture literature. The consistency of the DS maximum likelihood estimator is also established with any fixed smoothing parameter. Through numerical studies, we demonstrate that the proposed regression coefficient estimator has relatively good performance in terms of efficiency across a wide range of error distributions and robustness against outliers.