Abstract
The differential-geometrical framework is given for analyzing statistical problems related to multi-parameter families of distributions. The dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint. The duality connected by the Legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures. The second-order information loss is calculated for Fisher-efficient estimators, and is decomposed into the sum of two non-negative terms. One is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator. Only the latter term depends on the estimator, and vanishes for the maximum-likelihood estimator. A set of statistics which recover the second-order information loss are given. The second-order efficiency also is obtained. The differential geometry of the function space of distributions is discussed.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.