Location-adaptive density estimation and nearest-neighbor distance

A location-adaptive hybrid of the fixed-bandwidth kernel density estimate and the nearest-neighbor density estimate is introduced in this paper. It is constructed via a simple adhoc truncation and smoothing of nearest-neighbor distance. Simulations show that the hybrid outperforms its parent estimators, according to quadratic loss. Empirical process techniques are employed to obtain rates of uniform convergence of the random location-adaptive bandwidth to a deterministic function, from which uniform consistency of the hybrid, rates of convergence of the ISE, and asymptotic optimality of the ISE for the cross validatory choice of the smoothing parameter are obtained.