Some results about kernel estimators for function derivatives based on stationary and ergodic continuous time processes with applications

Abstract

The derivatives of the probability density or regression functions contain important information concerning a multivariate data set, such as modal regions. Despite this importance, nonparametric estimation of higher-order derivatives of the density or regression functions have received only relatively scant attention. The main purpose of the present work is to investigate general kernel type estimators of function derivatives. We obtain the strong uniform convergence with rate as well as the asymptotic normality for the proposed estimates. We consider the AMISE of kernel derivative estimator which plays a fundamental role for the characterization of the optimal bandwidth. Our results are obtained in the general setting of stationary ergodic processes. Finally, statistical applications include the regression derivatives, the multivariate mode, and the Shannon’s entropy, that are of independent interest.

Keywords:

• Conditional density
• kernel estimate
• Nadaraya-Watson estimators
• function derivatives
• ergodic processes
• martingale differences
• continuous time processes
• prediction

AMS 2010 SUBJECT CLASSIFICATIONS:

• 62G05
• 62G07
• 62G08
• 62E20
• 60F05
• 60E05
• 60A10

Acknowledgments

The authors are indebted to the Editor-in-Chief, Associate Editor, and the referees for their very valuable comments, suggestions careful reading of the article which led to a considerable improvement of the manuscript.

Author contributions

Salim Bouzebda and Sultana Didi: Conceptualization, Methodology, Investigation, Writing – Original Draft, and Writing – Review & Editing.