Central limit theorems for functional Z-estimators with functional nuisance parameters

Abstract

We consider an exchangeably weighted bootstrap for function-valued estimators defined as a zero point of a function-valued random criterion function. A large number of bootstrap resampling schemes emerge as special cases of our settings. The main ingredient is the use of a differential identity that applies when the random criterion function is linear in terms of the empirical measure. Our results are general and do not require linearity of the statistical model in terms of the unknown parameter. We also consider the semiparametric models extending Zhan's work to a more delicate framework. The theoretical results established in this paper are (or will be) key tools for further developments in the parametric and semiparametric models.

Keywords:

• Bootstrap
• Z-estimators
• estimating equations
• exchangeable
• infinite-dimensional
• nonparametric maximum likelihood

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F03
• 62F10
• 62F12
• 62H12
• 62H15

Acknowledgments

The authors are indebted to the Editor-in-Chief Professor Balakrishnan Narayanaswamy, Associate Editor and the two referees for their very generous comments and suggestions on the first version of our article which helped us to improve the content, presentation, and layout of the manuscript.