Bruce E. Hansen

"Exact Mean Integrated Squared Error of Higher-Order Kernels"

Econometric Theory (2005) 21, 1031-1057.


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) and the trapezoid kernel of Politis and Romano (1999), and are used to constrast their finite sample efficiency with the higher-order Gaussian kernels of Wand and Schucany (1990). 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 which 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 its characteristic function.

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Some of the above material is based upon work supported by the National Science Foundation under Grants No. SES-9022176, SES-9120576, SBR-9412339, and SBR-9807111. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s), and do not necessarily reflect the views of the NSF.