Adaptive sequential estimation for ergodic diffusion processes in quadratic metric

Abstract

An adaptive nonparametric procedure is constructed for estimating the unknown drift coefficient in ergodic diffusion processes. A sharp non-asymptotic upper bound (an oracle inequality) is obtained for a quadratic risk. Furthermore, an asymptotic lower bound for the minimax quadratic risk is found that equals to the Pinsker constant. Asymptotic efficiency is proved, that is, the asymptotic quadratic risk of the constructed estimator coincides with this constant.

Keywords:

• adaptive nonparametric estimation
• non-asymptotic estimation
• model selection
• sharp oracle inequality
• quadratic risk
• sequential estimation
• asymptotic bounds
• efficient estimation

AMS 2000 Subject Classifications :

• Primary: 62G08
• Secondary: 62G05
• 62G20

Acknowledgements

The authors are grateful to the reviewers for careful reading and valuable comments. S.M. Pergamenshchikov is partially supported by the RFBR-Grant 09-01-00172-a.