Computational Statistics

, Volume 18, Issue 2, pp 185–203

Multivariate Local Fitting with General Basis Functions

• Jochen Einbeck
Article

Summary

In this paper we combine the concepts of local smoothing and fitting with basis functions for multivariate predictor variables. We start with arbitrary basis functions and show that the asymptotic variance at interior points is independent of the choice of the basis. Moreover we calculate the asymptotic variance at boundary points. We are not able to compute the asymptotic bias since a Taylor theorem for arbitrary basis functions does not exist. For this reason we focus on basis functions without interactions and derive a Taylor theorem which covers this case. This theorem enables us to calculate the asymptotic bias for interior as well as for boundary points. We demonstrate how advantage can be taken of the idea of local fitting with general basis functions by means of a simulated data set, and also provide a data-driven tool to optimize the basis.

Key Words

Bias reduction local polynomial fitting multivariate kernel smoothing Taylor expansion

Notes

Acknowledgements

The author is grateful to Gerhard Tutz (LMU, Dep. of Statistics) for helpful inspirations during this work, Daniel Rost (LMU, Math. Inst.) for contributions concerning the extendibility of Taylor’s theorem and to the referees for many suggestions which improved this paper. This work was finished while the author passed a term at the University São Paulo. Many thanks especially to Carmen D. S. de André and Júlio M. Singer (USP) for their support in various fields.

Parts of this work were presented and discussed at the Euroworkshop on Statististical Modelling (2001). There a variety of valuable comments and suggestions were given that enhanced the paper.

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