On design-weighted local fitting and its relation to the Horvitz-Thompson estimator.

Weighting is a widely used concept in many fields of statistics and has frequently caused controversies on its justification and benefit. In this paper, we analyze design-weighted versions of the well-known local polynomial regression estimators, derive their asymptotic bias and variance, and observe that the asymptotically optimal weights are in conflict with (practically motivated) weighting schemes previously proposed in the literature. We investigate this conflict using theory and simulation, and find that the problem has a surprising counterpart in sampling theory, leading us back to the Discussion on Basu's (1971) elephants. In this light one might consider our result as an asymptotic and nonparametric version of the Horvitz-Thompson theorem. The crucial point is that bias-minimizing weights can make the estimators extremely vulnerable to outliers in the design space and have therefore to be used with particular care.