Differential geometry of edgeworth expansions in curved exponential family |
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Authors: | Shun-ichi Amari Masayuki Kumon |
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Institution: | (1) University of Tokyo, Tokyo, Japan |
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Abstract: | Summary In order to construct a higher-order asymptotic theory of statistical inference, it is useful to know the Edgeworth expansions
of the distributions of related statistics. Based on the differential-geometrical method, the Edgeworth expansions are performed
up to the third-order terms for the joint distribution of any efficient estimators and complementary (approximate) ancillary
statistics in the case of curved exponential family. The marginal and conditional distributions are also obtained. The roles
and meanings of geometrical quantities are elucidated by the geometrical interpretation of the Edgeworth expansions. The results
of the present paper provide an indispensable tool for constructing the differential-geometrical theory of statistics. |
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