Adaptive Nonparametric Peak Estimation
The Annals of Statistics · 1989 · Volume 17 · Issue 3
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.
Keywords
62G05, 62G20, choice of bandwidths, efficiency, Gaussian process, Kernel estimator, Nonparametric regression, size and location of peaks, tightness, weak convergence
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