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EXACT MEAN INTEGRATED SQUARED ERROR OF HIGHER ORDER KERNEL ESTIMATORS

Published online by Cambridge University Press:  23 September 2005

Bruce E. Hansen
Affiliation:
University of Wisconsin

Abstract

The exact mean integrated squared error (MISE) of the nonparametric kernel density estimator is derived for the asymptotically optimal smooth polynomial kernels of Müller (1984, Annals of Statistics 12, 766–774) and the trapezoid kernel of Politis and Romano (1999, Journal of Multivariate Analysis 68, 1–25) and is used to contrast their finite-sample efficiency with the higher order Gaussian kernels of Wand and Schucany (1990Canadian Journal of Statistics 18, 197–204). We find that these three kernels have similar finite-sample efficiency. Of greater importance is the choice of kernel order, as we find that kernel order can have a major impact on finite-sample MISE, even in small samples, but the optimal kernel order depends on the unknown density function. We propose selecting the kernel order by the criterion of minimax regret, where the regret (the loss relative to the infeasible optimum) is maximized over the class of two-component mixture-normal density functions. This minimax regret rule produces a kernel that is a function of sample size only and uniformly bounds the regret below 12% over this density class.

The paper also provides new analytic results for the smooth polynomial kernels, including their characteristic function.This research was supported in part by the National Science Foundation. I thank Oliver Linton and a referee for helpful comments and suggestions that improved the paper.

Type
Research Article
Copyright
© 2005 Cambridge University Press

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References

REFERENCES

Abadir, K.M. (1999) An introduction to hypergeometric functions for economists. Econometric Reviews 3, 287330.Google Scholar
Abadir, K.M. & S. Lawford (2004) Optimal asymmetric kernels. Economics Letters 83, 6168.Google Scholar
Chiu, S.-T. (1991) Bandwidth selection for kernel density estimation. Annals of Statistics 19, 18831905.Google Scholar
Davis, K.B. (1981) Mean integrated square error properties of density estimates. Annals of Statistics 5, 530535Google Scholar
Epanechnikov, V.I. (1969) Non-parametric estimation of a multivariate probability density. Theory of Probability and Its Applications 14, 153158.Google Scholar
Granovsky, B.L. & H.-G. Müller (1991) Optimizing kernel methods: A unifying variational principle. International Statistical Review 59, 373388.Google Scholar
Hall, P. & R.D. Murison (1993) Correcting the negativity of high-order kernel density estimators. Journal of Multivariate Analysis 47, 103122.Google Scholar
Hansen, B.E. (2003) Appendix: Exact mean integrated squared error of higher-order kernel estimators. www.ssc.wisc.edu/∼bhansen/papers/mise.html.
Magnus, W., F. Oberhettinger, & R.P. Soni (1966) Formulas and Theorems for the Special Functions of Mathematical Physics. Springer-Verlag.
Marron, J.S. & M.P. Wand (1992) Exact mean integrated squared error. Annals of Statistics 20, 712736.Google Scholar
Müller, H.-G. (1984) Smooth optimum kernel estimators of densities, regression curves and modes. Annals of Statistics 12, 766774.Google Scholar
Parzen, E. (1962) On estimation of a probability density and mode. Annals of Mathematical Statistics 35, 10651076.Google Scholar
Politis, D.N. & J.P. Romano (1999) Multivariate density estimation with general flat-top kernels of infinite order. Journal of Multivariate Analysis 68, 125.Google Scholar
Savage, L.J. (1951) The theory of statistical decision. Journal of the American Statistical Association 46, 5567.Google Scholar
Scott, D.W. (1992) Multivariate Density Estimation. Wiley.
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall.
Tranter, C.J. (1968) Bessel Functions with some Physical Applications. Hart.
Wand, M.P. & W.R. Schucany (1990) Gaussian-based kernels. Canadian Journal of Statistics 18, 197204.Google Scholar
Zhang, S. & J. Jin (1996) Computation of Special Functions. Wiley.