On the asymptotic normality of multistage integrated density derivatives kernel estimators

The estimation of integrated density derivatives is a crucial problem which arises in data-based methods for choosing the bandwidth of kernel and histogram estimators. In this paper, we establish the asymptotic normality of a multistage kernel estimator of such quantities, by showing that under some regularity conditions on the underlying density function and on the kernels used on the multistage estimation procedure, the multistage kernel estimator with at least one step of estimation is asymptotically equivalent in probability to the kernel estimator with associated optimal bandwidth. An application to kernel density bandwidth selection is also presented. In particular, we conclude that the common used plug-in bandwidth do not attempt the optimal rate of convergence to the optimal bandwidth.