Abstract: This paper studies boundary effects of the kernel density estimation and proposes some remedies to the problems. Since the kernel estimate is designed for estimating a smooth density, it introduces a large bias near the boundaries where the density is discontinuous. Bandwidth selectors developed for the kernel estimate that select a small bandwidth to reduce the bias can dramatically increase the variation and roughness of the density estimate. In this paper, several boundary adjusted procedures for estimating the density, as well as selecting the bandwidth, are introduced. The proposed procedures greatly reduce the boundary effects and is shown that these density estimates have the same optimal convergence rate as that of the kernel density estimate of a smooth density. Some asymptotic results about the boundary adjusted procedures are provided. Simulation studies were carried out to check the empiric performance of the proposed procedures compared to some existing boundary-corrected estimation procedures. In general, simulation results indicate that for moderate to large sample sizes, the proposed procedures reduce the boundary effects substantially, and are better than comparable existing methods. As an example, we estimate a relevant density connected with some coal-mining disaster data.
Key words and phrases: Bandwidth selection, boundary effects, characteristic function, cross-validation, kernel density estimation.