A note on the convergence rate of the kernel density estimator of the mode

In this paper, the mode estimator based on the Parzen-Rosenblatt kernel estimator is considered [Parzen, E., 1962. On estimating probability density function and mode. The Annals of Mathematical Statistics 33, 1065-1076]. In light of Shi et al. [Shi, X., Wu, Y., Miao, B., 2009. Strong convergence rate of estimators of change point and its application. Computational Statistics & Data Analysis 53, 990-998], under mild conditions, we establish the relationship between the convergence rate of the mode estimator and the window width. In this way, we obtain a better convergence rate of the mode estimator.