Data-driven bandwidth choice for gamma kernel estimates of density derivatives on the positive semi-axis

TL;DR

This paper develops data-driven methods for selecting optimal bandwidths in gamma kernel density derivative estimation on the positive semi-axis, improving estimation accuracy for applications involving positive data.

Contribution

It introduces explicit formulas for optimal bandwidths and constructs practical data-driven rules like 'rule of thumb' and 'cross-validation' for gamma kernel derivatives.

Findings

Bandwidth selection significantly improves estimation accuracy.

Data-driven methods outperform fixed bandwidth choices.

Demonstrated effectiveness on Maxwell and Weibull distributions.

Abstract

In some applications it is necessary to estimate derivatives of probability densities defined on the positive semi-axis. The quality of nonparametric estimates of the probability densities and their derivatives are strongly influenced by smoothing parameters (bandwidths). In this paper an expression for the optimal smoothing parameter of the gamma kernel estimate of the density derivative is obtained. For this parameter data-driven estimates based on methods called "rule of thumb" and "cross-validation" are constructed. The quality of the estimates is verified and demonstrated on examples of density derivatives generated by Maxwell and Weibull distributions.