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

Einbeck, Jochen; Augustin, Thomas

Abstract

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.

Citation

Einbeck, J., & Augustin, T. (2007). On design-weighted local fitting and its relation to the Horvitz-Thompson estimator. Statistica sinica, 19(1), 103-123

• Journal Article Type: Article
• Online Publication Date: Mar 15, 2007
• Publication Date: Mar 15, 2007
• Deposit Date: Jan 7, 2009
• Journal: Statistica Sinica
• Print ISSN: 1017-0405
• Electronic ISSN: 1996-8507
• Publisher: Institute of Statistical Science, Academia Sinica
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 1
• Pages: 103-123
• Keywords: Bias reduction, Nonparametric smoothing, Local polynomial modelling, Kernel smoothing, Leverage values, Horvitz-Thompson theorem, Stratification.
• Public URL: https://durham-repository.worktribe.com/output/1546174
• Publisher URL: http://www3.stat.sinica.edu.tw/statistica/j19n1/j19n15/j19n15.html