Adaptive Nonparametric Peak Estimation

Abstract

It is shown that consistent estimates of the optimal bandwidths for kernel estimators of location and size of a peak of a regression function are available. Such estimates yield the same joint asymptotic distribution of location and size of a peak as the optimal bandwidths themselves. Therefore data-adaptive efficient estimation of peaks is possible. In order to prove this result, the weak convergence of a two-dimensional stochastic process with appropriately scaled bandwidths as arguments to a Gaussian limiting process is shown. A practical method which leads to consistent estimates of the optimal bandwidths and is therefore asymptotically efficient is proposed and its finite sample properties are investigated by simulation.