Generalised local polynomial estimators of smooth functionals of a distribution function with nonnegative support

Abstract

This paper introduces generalised smooth asymmetric kernel estimators for smooth functionals with non-negative support. More precisely, for $x\in [0,\mathrm{\infty }),$ and for a functional, $\mathrm{\Phi }(x,F)$ of the distribution function F, we develop estimators of the functional Φ and its derivatives. The proposed estimator can be seen as the solution to a minimisation problem in the polynomial space $L^2\left(q\right),$ where q is an asymmetric density function. The framework presented here covers several classical nonparametric functional estimators and is linked with estimation using hierarchical kernels. We establish the asymptotic properties of the proposed estimators in the general framework. Furthermore, special attention is paid to comparing the asymptotic mean integrated square error (AMISE) of the proposed estimator with that of other classical symmetric/asymmetric density estimators. Additionally, a comparison of finite sample behaviour is conducted for both density estimation and hazard rate estimation via simulation and real data application.

Keywords:

• Asymmetric kernels
• bounded support
• density estimation
• hazard rate estimate
• hierarchical kernels
• local polynomial fitting
• smooth functionals estimators

AMS 2000 Subject Classifications:

• Primary: 62G05
• Secondary: 62G07
• 62G20

Acknowledgements

We thank the editor, associate editor and the two referees for their valuable and constructive comments which helped improve the manuscript substantially.

Disclosure statement

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

Additional information

Funding

This work received funding from CY Advanced Studies of CY Cergy Paris University and was partly developed during the second author's visit to CY Advanced Studies. The first author would like to acknowledge the partial support for this research from NSERC of Canada through a Discovery Grant.