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Adaptive nonparametric peak estimation. (English) Zbl 0683.62019
Summary: It is shown that consistent estimates of the optimal bandwidths for kernel estimators of location and size of 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.

MSC:
62G05 Nonparametric estimation
60F05 Central limit and other weak theorems
62G20 Asymptotic properties of nonparametric inference
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