ON LARGE DEVIATION EXPANSION OF DISTRIBUTION OF MAXIMUM LIKELIHOOD ESTIMATOR AND ITS APPLICATION IN LARGE SAMPLE ESTIMATION

J. C. FU^{1}, GANG LI^{2} AND L. C. ZHAO^{3}

1Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 2Department of Math. Sciences, SUNY at Binghamton, Binghamton, NY 13902, U.S.A.

3Department of Mathematics, Univ. of Science and Technology of China, Hefie, Anhui, China

(Received April 10, 1991; revised September 29, 1992)

Abstract. For estimating an unknown parameter θ, the likelihood principle yields the maximum likelihood estimator. It is often favoured especially by the applied statistician, for its good properties in the large sample case. In this paper, a large deviation expansion for the distribution of the maximum likelihood estimator is obtained. The asymptotic expansion provides a useful tool to approximate the tail probability of the maximum likelihood estimator and to make statistical inference. Theoretical and numerical examples are given. Numerical results show that the large deviation approximation performs much better than the classical normal approximation.

Key words and phrases: Large deviation expansion, maximum likelihood estimator, exponential rate.