Consistency of the Subsample Bootstrap empirical process

Abstract

In the classical Bootstrap approach the number of distinct observation in the resample is random. To overcome this hitch Rao et al. [Bootstrap by sequential resampling, J. Statist. Plan. Inference 64 (1997), pp. 257–281] have proposed a modified resampling procedure – the so-called Sequential Bootstrap or 0.632-Bootstrap – in which each resample has exactly the same number m ∼eq ⌊0.632 n⌋ of distinct observations. Motivated by this idea we introduce an akin procedure, the Subsample Bootstrap, where additionally even the size of each resample is equal. It will turn out that the Subsample Bootstrap empirical process is consistent for a wide class of Donsker classes.

Keywords:

• bootstrap
• sequential bootstrap
• empirical process
• Donsker classes
• resampling methods

AMS Subject Classification :

• 62G09
• 62L10

Acknowledgements

We are especially grateful to the Editor, Associate Editor and two anonymous referees in terms of their careful reading of our manuscript, which led to an improved presentation.

Additional information

Notes on contributors

Markus Pauly

Parts of this article have been developed while the author was a visiting scholar at the Statistics Department at the University of Washington, Seattle.