Regression using localised functional Bregman divergence

Abstract

This paper is concerned with a unified approach to estimating regression methods based on a certain divergence and its localisation. Some past papers have demonstrated theoretically and numerically that infusing a little localisation in the likelihood-based methods for regression and for density estimation can actually improve the resulting estimators with respect to suitably defined global risk measures. Thus a variety of local likelihood methods have been suggested. We demonstrate that similar effect can also be observed in the general framework discussed in this paper and with respect to robust estimation procedures. Localised versions of robust regression estimation procedures perform better with respect to global risk measures based on minimisation of Bregman divergence measures. An intricate relationship between regression model’s inadequacy and its robustness can be better analysed by using the local approach developed in this paper. We support our claims with a short simulation study.