Riemannian manifold of multivariate distributions in view of geometrical aspects

2017 Fiscal Year Final Research Report

• Project/Area Number: 25380265
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Economic statistics
• Research Institution: Shinshu University
• Principal Investigator: Shiina Yo 信州大学, 学術研究院社会科学系, 教授 (80242709)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 情報幾何 / リーマン多様体 / ダイバージェンス / 曲率 / 接続

Outline of Final Research Achievements

1) Consider a family of parametric distributions. We get a predictive distribution by inserting the maximum likelihood estimator. We measure the discrepancy the predictive distribution and the true distribution by alpha-divergence and define the risk as its expectation. We derived an asymptotic expansion w.r.t. the sample size. The first term ,i.e. 1/n term equals (the number of the parameters)/2 and the second term, i.e. 1/n^2 term is expressed with various geometrical properties. We gained concrete results for a multinomial distribution, a multivariate normal distribution, a mixture family. 2) We applied the above general result to a multiple regression model with a parametric error distribution. We investigated how the distribution of the explanatory variables and the error distribution affect the geometrical properties in the second term.

Free Research Field

数理統計学