ON SOME OPTIMALITY PROPERTIES OF FISHER-RAO SCORE FUNCTION IN TESTING AND ESTIMATION

Abstract

The score function is associated with some optimality features in statistical inference. This review article looks on the central role of the score in testing and estimation. The maximization of the power in testing and the quest for efficiency in estimation lead to score as a guiding principle. In hypothesis testing, the locally most powerful test statistic is the score test or a transformation of it. In estimation, the optimal estimating function is the score. The same link can be made in the case of nuisance parameters: the optimal test function should have maximum correlation with the score of the parameter of primary interest. We complement this result by showing that the same criterion should be satisfied in the estimation problem as well.

Keywords:

• Score function
• Rao's score test
• Locally most powerful test
• Estimating function
• Efficiency
• Efficient method of moment

ACKNOWLEDGMENTS

This paper is based on an invited presentation at the international conference, Statistics: Reflections on the Past and Visions for the Futurein honor of Professor C. R. Rao on the occasion of his 80th birthday, held at the University of Texas at San Antonio, March 16–19, 2000. We wish to thank the participants for helpful comments and discussion. We are grateful to an anonymous referee for many helpful suggestions that have considerably improved the paper. We would also like to thank Tatyana Dubovyk for providing very competent research assistance and to Naoko Miki for her help in preparing the manuscript. However, we retain the responsibility for any remaining errors. Financial support from the Office of Research, College of Commerce and Business Administration, University of Illinois of Urbana-Champaign is gratefully acknowledged.