Robust-BD Estimation and Inference for General Partially Linear Models

The classical quadratic loss for the partially linear model (PLM) and the likelihood function for the generalized PLM are not resistant to outliers. This inspires us to propose a class of “robust-Bregman divergence (BD)” estimators of both the parametric and nonparametric components in the general partially linear model (GPLM), which allows the distribution of the response variable to be partially specified, without being fully known. Using the local-polynomial function estimation method, we propose a computationally-efficient procedure for obtaining “robust-BD” estimators and establish the consistency and asymptotic normality of the “robust-BD” estimator of the parametric component βo. For inference procedures of βo in the GPLM, we show that the Wald-type test statistic Wn constructed from the “robust-BD” estimators is asymptotically distribution free under the null, whereas the likelihood ratio-type test statistic Λn is not. This provides an insight into the distinction from the asymptotic equivalence (Fan and Huang 2005) between Wn and Λn in the PLM constructed from profile least-squares estimators using the non-robust quadratic loss. Numerical examples illustrate the computational effectiveness of the proposed “robust-BD” estimators and robust Wald-type test in the appearance of outlying observations.