Computational Statistics

, Volume 18, Issue 2, pp 185–203 | Cite as

Multivariate Local Fitting with General Basis Functions

  • Jochen Einbeck


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 



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.


  1. Cleveland, W. S. (1979). Robust locally weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc. 74, 829–836.MathSciNetCrossRefGoogle Scholar
  2. Cleveland, W. S. and Devlin, S. (1988). Locally weighted regression: An approach to regression analysis by local fitting. J. Amer. Statist. Assoc. 83, 596–610.CrossRefGoogle Scholar
  3. Einbeck, J. (2001). Local fitting with general basis functions, SFB 386, Discussion Paper No. 256.∼einbeck/
  4. Fahrmeir, L. and Hamerle, A. (1984). Multivariate statistische Verfahren. Berlin / New York: de Gruyter.zbMATHGoogle Scholar
  5. Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87, 998–1004.MathSciNetCrossRefGoogle Scholar
  6. Nadaraya, E. A. (1964). On estimating regression. Theory Prob. Appl. 10, 186–190.CrossRefGoogle Scholar
  7. Ramsay, J. O. and Silverman, B. W. (1997). Functional Data Analysis. New York: Springer.CrossRefGoogle Scholar
  8. Ruppert, D. and Wand, M. P. (1994). Multivariate locally weighted least squares regression. Ann. Statist. 22, 1346–1370.MathSciNetCrossRefGoogle Scholar
  9. Staniswalis, J. G., Messer, K., and Finston, D. R. (1993). Kernel estimators for multivariate regression. Nonparametric Statistics 3, 103–121.MathSciNetCrossRefGoogle Scholar
  10. Stone, C. J. (1977). Consistent nonparametric regression. Ann. Statist. 5, 595–645.MathSciNetCrossRefGoogle Scholar
  11. Wand, M. P. (1992). Error analysis for general multivariate kernel estimators. Nonparametric Statistics 2, 1–15.MathSciNetCrossRefGoogle Scholar
  12. Wand, M. P. and Jones, M. C. (1993). Comparison of smoothing parametrizations in bivariate kernel density estimation. J. Amer. Statist. Assoc. 88, 520–528.MathSciNetCrossRefGoogle Scholar
  13. Watson, G. S. (1964). Smooth regression analysis. Sankhyā, Series A, 26, 359–372.MathSciNetzbMATHGoogle Scholar
  14. Yang, L. and Tschering, R. (1999). Multivariate bandwidth selection for local linear regression. J. R. Statist. Soc. B 61, 793–815.MathSciNetCrossRefGoogle Scholar

Copyright information

© Physica-Verlag 2003

Authors and Affiliations

  • Jochen Einbeck
    • 1
  1. 1.Institut für StatistikLudwig Maximilians UniversitätMünchenGermany

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