Efficiency of a Class of Unbiased Estimators for the Invariant Distribution Function of a Diffusion Process

Abstract

We consider the problem of the estimation of the invariant distribution function of an ergodic diffusion process when the drift coefficient is unknown. The empirical distribution function is a natural estimator which is unbiased, uniformly consistent and efficient in different metrics. Here we study the properties of optimality for another kind of estimator recently proposed. We consider a class of unbiased estimators and we show that they are also efficient in the sense that their asymptotic risk, defined as the integrated mean square error, attains the same asymptotic minimax lower bound of the empirical distribution function.

Keywords:

• Asymptotically efficient estimators
• Ergodic diffusion
• Lower bound

Mathematics Subject Classification:

• 60G35
• 62M20

Acknowledgments

My warmest thanks go to Yury A. Kutoyants for his useful advices and clarifying discussions.

This work has been partially supported by the local grant sponsored by the University of Bergamo: Theoretical and computational problems in statistics for continuously and discretely observed diffusion processes and MIUR 2004 Grant.