On optimal estimation of a non-smooth mode in a nonparametric regression model with α-mixing errors

We consider the problem of mode estimation in the fixed-design regression model, the regression function having a unique non-smooth mode. We estimate the mode by maximization over the curve estimator, which is given as a weighted mean of the observations, including most of the common kernel estimators, such as Gasser-Müller, Priestley-Chao and Nadaraya-Watson. To obtain optimal rates of convergence of the mode estimator, we first derive upper bounds, where we benefit from the contrast of the curve at the mode rather than taking into account the rate of uniform convergence of the curve estimator. In a next step we show that these rates are also optimal. We prove our results for α-mixing observations, and a non-smooth regression function that is only assumed to be Hölder continuous. Our method consists in a rather direct evaluation of the mean squared error of the empirical mode, using a recent moment inequality of Rosenthal type due to Yang [2007. Maximal moment inequality for partial sums of strong mixing sequences and application. Acta Math. Sinica 23, 1013-1024] for mixing random variables.