Some Asymptotic Results of Kernel Density Estimators Under Random Left-Truncation and Dependent Data

Abstract

Problems with truncated data arise frequently in survival analyses and reliability applications. The estimation of the density function of the lifetimes is often of interest. In this article, the estimation of density function by the kernel method is considered, when truncated data are showing some kind of dependence. We apply the strong Gaussian approximation technique to study the strong uniform consistency for kernel estimators of the density function under a truncated dependent model. We also apply the strong approximation results to study the integrated square error properties of the kernel density estimators under the truncated dependent scheme.

Keywords:

• Integrated square error
• Kiefer process
• Strong Gaussian approximation
• Strong mixing
• Strong uniform consistency
• Truncated data

Mathematics Subject Classification:

• 62G07
• 62G20

Acknowledgment

The authors would like to sincerely thank the anonymous referee for his/her valuable comments. The first author was supported by a grant from Ferdowsi Universtiy of Mashhad (No. MS89138FAK).