Geometry of exponential family nonlinear models and some asymptotic inference (Q1895489)

Exponential family nonlinear models include many important statistical models, such as generalized linear models and nonlinear regression models. This paper presents a differential geometric framework for these models. Based on this framework, some asymptotics related to curvatures are studied. \textit{S. Amari} [see ``Differential-geometrical methods in statistics''. (1985; Zbl 0559.62001)] developed the differential geometric method of statistics presented first by \textit{B. Efron} [Ann. Stat. 3, No. 6, 1189-1242 (1975; Zbl 0321.62013)] by introducing a Riemannian geometric framework for curved exponential families. On the other hand, \textit{D. M. Bates} and \textit{D. G. Watts} [J. R. Stat. Soc., Ser. B 42, 1-25 (1980; Zbl 0455.62028)] introduced the curvature measures for nonlinear regression models in Euclidean space. Because the Euclidean geometry is relatively simple and the regression model is quite useful, many authors have concentrated on this approach. Comparing the geometries of Efron-Amari and Bates-Watts, the former can be applied to a quite general class of families, but the geometry based on the Riemannian manifold might be rather complicated; the latter is relatively simple and intuitive, but it can be only used for nonlinear regression models. We try to combine the advantages of both Efron-Amari and Bates-Watts geometries and present a differential geometric framework in Euclidean space for quite general models, the exponential family nonlinear models. Briefly speaking, the geometry we use is analogous to that of the Bates-Watts framework, but we introduce a Fisher information inner product that Bates-Watts have not considered; the distribution family we study is analogous to that of the Efron-Amari framework, but they studied the classical iid case and we study the non-iid regression case.