Weighted bootstrapped kernel density estimators in two-sample problems

ABSTRACT

A weighted bootstrap method is proposed to approximate the distribution of the () norms of two-sample statistics involving kernel density estimators. Using an approximation theorem of Horváth, Kozkoszka and Steineback [(2000) ‘Approximations for Weighted Bootstrap Processes with an Application’, Statistics and Probability Letters, 48, 59–70], that allows one to replace the weighted bootstrap empirical process by a sequence of Gaussian processes, we establish an unconditional bootstrap central limit theorem for such statistics. The proposed method is quite straightforward to implement in practice. Furthermore, through some simulation studies, it will be shown that, depending on the weights chosen, the proposed weighted bootstrap approximation can sometimes outperform both the classical large-sample theory as well as Efron's [(1979) ‘Bootstrap Methods: Another Look at the Jackknife’, Annals of Statistics, 7, 1–26] original bootstrap algorithm.

KEYWORDS:

• Kernel
• density
• weighted bootstrap
• CLT
• Brownian bridge

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G07
• 62G08
• 62G09

Acknowledgments

The authors would like to thank Editor, Iréne Gijbels, Associate Editor, and anonymous referees for their comments on the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the National Science Foundation under Grant DMS-1407400 of Majid Mojirsheibani.