On the Asymptotic Distributions of Bandwidth Estimates

Abstract

The problem of automatic bandwidth selection for a kernel regression estimate is studied. Since the bandwidth considerably affects the features of the estimated curve, it is important to understand the behavior of bandwidth selection procedures. The bandwidth estimate considered here is the minimizer of Mallows' criterion. Though it was established that the bandwidth estimate is asymptotically normal, it is well recognized that the rate of convergence is extremely slow. In simulation studies, it is often observed that the normal distribution does not provide a satisfactory approximation. In this paper, the bandwidth estimate is shown to be approximately equal to a constant plus a linear combination of independent exponential random variables. In practice, the distribution of a weighted sum of chi-squared random variables can be approximated by a multiple of a chi-squared distribution. Simulation results indicate that this provides a very good approximation even for a modest sample size. It is shown that the degrees of freedom of the chi-squared distribution goes to infinity rather slowly (at the rate $T^{1/5}$ for a nonnegative kernel). This explains why the distributions of the bandwidth estimates converge to the normal distribution so slowly.