Density derivative estimation using asymmetric kernels

Abstract

This paper studies the problem of estimating the first-order derivative of an unknown density with support on ${\mathbb{R}}_{+}$ or $[0,1]$. 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 $n^{-4/7}$ 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.

Keywords:

• Beta kernel
• density derivative estimation
• digamma function
• gamma kernel
• trigamma function

JEL Classification Codes:

• C13
• C14

MSCs 2010:

• 62G08
• 62G20

Acknowledgments

The authors would like to thank the editor and two anonymous referees for their constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability

The datasets used in Section 4 are openly available at the following URLs:

• http://qed.econ.queensu.ca/jae/1998-v13.2/vella-verbeek/ (US hourly wages)

• https://www.stat.cmu.edu/∼larry/all-of-nonpar/data.html (eruption durations of the Old Faithful Geyser).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science under grant numbers 19K01595 and 23K01340 (M. Hirukawa).