Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions
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 typical smoothness conditions. Only the uniform rate of the co-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 non-smooth) the mode is, while the contrary holds with estimation of the size of the mode.
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