Asymptotic properties of the kernel mode estimator under twice censorship model

ABSTRACT

In this article, we study the asymptotic properties of the kernel estimator of the mode and density function when the data are twice censored. More specifically, we first establish a strong uniform consistency over a compact set with a rate of the kernel density estimator and then we give the consistency with rate and asymptotic normality for the kernel mode estimator. An application to confidence bands is given.

KEYWORDS:

• Asymptotic normality
• Kernel mode estimation
• Patilea–Rolin estimator
• Rate of convergence
• Strong uniform consistency
• Twice censoring

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05
• 62G07
• 62G20

Acknowledgments

The authors are grateful to the anonymous reviewer for his/her particularly careful reading, relevant remarks, and constructive comments, which helped the authors to improve the quality and the presentation of an earlier version of this article.

Notes

1 A deeper investigation of the theoretical and practical point of view of optimal bandwidth choice is beyond the scope of the present article and postponed to future research.