Exact mean integrated squared error of higher order kernel estimators

The exact mean integrated squared error (MISE) of the nonparametric kernel density estimator is derived for the asymptotically optimal smooth polynomial kernels of Muller (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 (1990 Canadian 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.