ESTIMATES OF DERIVATIVES OF (LOG) DENSITIES AND RELATED OBJECTS

Abstract

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.