Estimates of derivatives of (log) densities and related objects

TL;DR

This paper introduces a nonnegative density and derivative estimator based on local polynomial approximation of the log-density, achieving optimal convergence rates and outperforming kernel methods in simulations.

Contribution

The paper proposes a novel nonnegative density derivative estimator using local polynomial log-density approximation with optimal convergence and practical advantages.

Findings

Estimator is nonnegative and achieves optimal convergence rates.

Performs favorably compared to kernel methods in simulations.

Can outperform optimized kernel methods with larger bandwidths.

Abstract

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density $f$. The estimator is guaranteed to be nonnegative and achieves the same optimal rate of convergence in the interior as well as the boundary of the support of $f$. The estimator is therefore well-suited to applications in which nonnegative 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 in finite samples to these alternative methods when they are used with optimal inputs, i.e. an Epanechnikov kernel and optimally chosen bandwidth sequence. Further simulation evidence demonstrates that, if the researcher modifies the inputs and chooses a larger bandwidth, our approach can even improve upon these optimized alternatives, asymptotically. We provide code in several languages.