Optimal plug-in estimators for multivariate distributions with conditionally independent components

Abstract

The usual estimator for the expectation of a function of a random vector is the empirical estimator. Assume that some of the components of the random vector are conditionally independent given the other components. We construct a plug-in estimator for the expectation that uses this information, prove a central limit theorem for the estimator, and show that the estimator is asymptotically efficient in the sense of a nonparametric version of the convolution theorem of Hájek and Le Cam.

Keywords:

• smoothed empirical estimator
• multivariate density estimator
• local asymptotic normality

2010 AMS Subject Classifications :

• Primary: 62G05
• Secondary: 62G20

Acknowledgements

Ursula U. Müller was supported by NSF Grant DMS 0907014. Anton Schick was supported by NSF Grant DMS 0906551. The authors thank the referees and an Associate Editor for a number of suggestions that improved the manuscript.