Kernel Estimation of Densities with Discontinuities or Discontinuous Derivatives

In kernel density estimation, one usually assumes the density has two continous derivatives. In this paper we give precise experssions for the asymptotic mean integrated squared error in case the density has m-1 continuous derivatives and two more derivatives with simple discontinuities. We show that the convergence rate for the mean integrated squared error, with appropriate kernel, is proportional to n-v,ν =2m+1/2m+2. Furthermore the ratio of the (random) integrated squared error to the mean integrated squared error converges to 1 almost surely, if the bandwidth is chosen by cross-validation. In particular, it is possible to have n-5/6 with only one continuous derivative, if the kernel and bandwidth are appropriately chosen. When the density has known points of discontinuity, a symmetrization device suggested by SCHUSTER (1985) can be used to improve the convergence rate. The mean integrated squared error of the symmetrization estimator will converge as if the density had no discontinuities at those points. Typically, the rate will be proportional to n-3/4