Convergence Rate of Strong Consistency of the Maximum Likelihood Estimator in Exponential Family Nonlinear Models

Abstract

This article proposes some regularity conditions. On the basis of the proposed regularity conditions, we show the strong consistency of the maximum likelihood estimator (MLE) in exponential family nonlinear models (EFNM) and give its convergence rate. In an important case, we obtain the convergence rate O(n ^−1/2(log log n)^1/2)—the rate as that in the Law of the Iterated Logarithm (LIL) for iid partial sums and thus cannot be improved anymore.

Keywords:

• Convergence rate
• Exponential family nonlinear models
• Maximum likelihood estimator
• Strong consistency

Mathematics Subject Classification:

• Primary 62F12, 62G02
• Secondary 62G12

Acknowledgments

We would like to thank the referee for helpful comments. We gratefully acknowledge financial support from Natural Science Foundation of Yunnan University (2005Z007C) and Natural Science Foundation of Educational Department of Yunnan Province (5Y0062A).