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Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions. (English) Zbl 1086.62046

Summary: Nonparametric estimation of the mode of a density or regression function via kernel methods is considered. It is shown that the rate of consistency of the mode estimator can be determined without the typical smoothness conditions. Only the uniform rate of the so-called stochastic part of the problem together with some mild conditions characterizing the shape or “acuteness” of the mode influence the rate of the mode estimator. In particular, outside the location of the mode, our assumptions do not even imply continuity. Overall, it turns out that the location of the mode can be estimated at a rate that is the better the “peakier” (and hence nonsmooth) the mode is, while the contrary holds with estimation of the size of the mode.

MSC:

62G07 Density estimation
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
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