Smooth simultaneous confidence bands for cumulative distribution functions

Abstract

A plug-in kernel estimator is proposed for Hölder continuous cumulative distribution function (cdf) based on a random sample. Uniform closeness between the proposed estimator and the empirical cdf estimator is established, while the proposed estimator is smooth instead of a step function. A smooth simultaneous confidence band is constructed based on the smooth distribution estimator and the Kolmogorov distribution. Extensive simulation study using two different automatic bandwidths confirms the theoretical findings.

Keywords:

• bandwidth
• confidence band
• Hölder continuity
• kernel
• Kolmogorov distribution

AMS Subject Classifications :

• 62G08
• 62G10
• 62G20

Acknowledgements

This research has been supported in part by NSF Award DMS 1007594, Jiangsu Specially-Appointed Professor Program and Jiangsu Key Discipline Program (Statistics), Jiangsu, China. The helpful comments by the Editor-in-Chief and two referees are gratefully acknowledged.