Curved Exponential Families and Edgeworth Expansions

Abstract

Part II is devoted to the higher-order asymptotic theory of statistical inference in curved exponential family M imbedded in exponential family S. A number of independent observations are summarized into a vector sufficient statistics $\bar x$ in a curved exponential family, which defines an observed point or distribution in S. In chapter, we decompose $\bar x$ into a pair (û, v) of statistics such that û is asymptotically sufficient and v is asymptotically ancillary. The Edeworth expansion of the joint distribution p (û, v) is given explicitly up to the third order terms by using the related geometrical quantities in S and M.