Density derivative estimation using asymmetric kernels

This paper studies the problem of estimating the first-order derivative of an unknown density with support on $ \mathbb {R}_{+} $ R+ or $ \left [0,1\right ] $ [0,1]. Nonparametric density derivative estimators smoothed by the asymmetric, gamma and beta kernels are defined, and their convergence properties are explored. It is demonstrated that these estimators can attain the optimal convergence rate of the mean integrated squared error $ n^{-4/7} $ n−4/7 when the underlying density has third-order smoothness. Superior finite-sample properties of the proposed estimators are confirmed in Monte Carlo simulations, and usefulness of the estimators is illustrated in two real data examples.