A new weighted likelihood approach

In this paper, we propose a new weighted likelihood procedure. Here, the weights are suitably calibrated functions of appropriately described residuals at each data point. The residuals describe the match (or mismatch) between the empirical distribution function and the model distribution function. If the match is high, the observation is considered to be a regular observation. But for large (in magnitude) residuals, there is a mismatch, and the corresponding likelihood score function may require downweighting in order to obtain a robust solution. As there is little or no downweighting for observations where there is no evidence of mismatch, asymptotically, we expect that there will be no downweighting under the pure model leading to highly efficient estimators. On the other hand, properly calibrated weight functions that penalize the observations with large residuals will lead to highly robust solutions under model misspecification and the presence of outliers. Copyright © 2015 John Wiley & Sons, Ltd.