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

Weighting is a largely used concept in many fields of statistics and has frequently cause controversies on its justification and profit. In this paper, we analyze a weighted version of the well-known local polynomial regression estimators, derive their asymptotic bias and variance, and find that the conflict between the asymptotically optimal weighting scheme and the practical requirements has a surprising counterpart in sampling theory, leading us back to the discussion on Basu's (1971) elephants, -- Bias reduction ; nonparametric smoothing ; local polynomial modelling ; kernel smoothing : leverage values ; Horvitz-Thompson theorem ; stratification