The law of the iterated logarithm and maximal smoothing principle for the kernel distribution function estimator

Abstract

Two new properties of the kernel distribution function estimator of diverse nature are derived. Firstly, a law of the iterated logarithm is proved for both the integrated absolute error and the integrated squared error of the estimator. Secondly, the maximal smoothing principle in kernel density estimation developed by Terrell is extended to kernel distribution function estimation, which allows, among others, the derivation of an alternative quick-and-simple bandwidth selector. In fact, there is a common link between the two topics: both problems are solved through the use of the same, not-so-standard, methodology. The results based on simulated data and a real data set are also presented.

Keywords:

• Bandwidth
• kernel distribution function estimator
• law of the iterated logarithm
• maximal smoothing principle
• smoothed bootstrap

2010 Mathematics Subject classifications:

• Primary 62G05
• 62G20
• 60F15
• Secondary60F17
• 62F40

Acknowledgements

The author would like to thank the referees for suggestions that improved this paper.

Disclosure statement

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