Differential-Geometrical Methods in Statistics

1. Introduction.- I. Geometrical Structures of a Family of Probability Distributions.- 2. Differential Geometry of Statistical Models.- 2.1. Manifold of statistical model.- 2.2. Tangent space.- 2.3. Riemannian metric and Fisher information.- 2.4. Affine connection.- 2.5. Statistical a-connection.- 2.6. Curvature and torsion.- 2.7. Imbedding and submanifold.- 2.8. Family of ancillary submanifolds.- 2.9. Notes.- 3. ?-Divergence and ?-Projection in Statistical Manifold.- 3.1. ?-representation.- 3.2. Dual affine connections.- 3.3. ?-family of distributions.- 3.4. Duality in ?-flat manifolds.- 3.5. ?-divergence.- 3.6. ?-projection.- 3.7. On geometry of function space of distributions.- 3.8. Remarks on possible divergence, metric and connection in statistical manifold.- 3.9. Notes.- II. Higher-Order Asymptotic Theory of Statistical Inference in Curved Exponential Families.- 4. Curved Exponential Families and Edgeworth Expansions.- 4.1. Exponential family.- 4.2 Curved exponential family.- 4.3. Geometrical aspects of statistical inference.- 4.4. Edgeworth expansion.- 4.5. Notes.- 5. Asymptotic Theory of Estimation.- 5.1. Consistency and efficiency of estimators.- 5.2. Second- and third-order efficient estimator.- 5.3. Third-order error of estimator without bias correction.- 5.4. Ancillary family depending on the number of observations.- 5.5. Effects of parametrization.- 5.6. Geometrical aspects of jackknifing.- 5.7. Notes.- 6. Asymptotic Theory of Tests and Interval Estimators.- 6.1. Ancillary family associated with a test.- 6.2. Asymptotic evaluations of tests: scalar parameter case.- 6.3. Characteristics of widely used efficient tests: Scalar parameter case.- 6.4. Conditional test.- 6.5. Asymptotic properties of interval estimators.- 6.6. Asymptotic evaluations of tests: general case.- 6.6. Notes.- 7. Information, Ancillarity and Conditional Inference.- 7.1. Conditional information, asymptotic sufficiency and asymptotic ancillarity.- 7.2. Conditional inference.- 7.3.