Entropy, Vol. 19, Pages 625: Robust-BD Estimation and Inference for General Partially Linear Models (Entropy)

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 ? o . For inference procedures of ? o in the GPLM , we show that the Wald-type test statistic W n constructed from the `robust- BD ` estimators is asymptotically distribution free under the null, whereas the likelihood ratio-type test statistic ? n is not. This provides an insight into the distinction from the asymptotic equivalence (Fan and Huang 2005) between W n and ? n 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.