Sankhya: The Indian Journal of Statistics

1997, Volume 59, Series A, Pt. 2, 179-197

BEYOND THIRD-ORDER EFFICIENCY

By

YUTAKA KANO, University of Tsukuba, Ibaraki

SUMMARY. Fifth-order (asymptotic) efficiency of the second-order bias-corrected MLE, minimizing the n -3 term of an expansion of the quadratic risk Fisher-consistent estimators bias-corrected similarly, is established in a general curved exponential family with a structural parameter vector. A characterization theorem of the MLE in terms of its higher -order derivatives is provided, and an alternative bias correction factor is proposed. Both the characterization and the new bias play an important role in proving the fifth-order derivatives are utilized, rather than usual element wise tensors with Einestein's convention, to derive all the results of this article.

AMS (1991) subject classification. 62F12.

Key words and phrases. Curved exponential family, Fisher-consistency, higher-order matrix derivatives, maximum likelihood estimator, quadratic risk, symmetric tensor.

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