Differential-Geometrical Methods in Statistics

ISBN-13: 9780387960562 / Angielski / Miękka / 1990 / 294 str.

Shun'ichi Amari; Shun-Ichi Amari

Opis

From the reviews: "In this Lecture Note volume the author describes his differential-geometric approach to parametrical statistical problems summarizing the results he had published in a series of papers in the last five years. The author provides a geometric framework for a special class of test and estimation procedures for curved exponential families. ... ... The material and ideas presented in this volume are important and it is recommended to everybody interested in the connection between statistics and geometry ..." #Metrika#1 "More than hundred references are given showing the growing interest in differential geometry with respect to statistics. The book can only strongly be recommended to a geodesist since it offers many new insights into statistics on a familiar ground." #Manuscripta Geodaetica#2

Spis treści

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. Pooling independent observations.- 7.4. Complete decomposition of information.- 7.5. Notes.- 8. Statistical Inference in the Presence of Nuisance Parameters.- 8.1. Orthogonal parametrization and orthogonalized information.- 8.2. Higher-order efficiency of estimators.- 8.3. The amount of information carried by knowledge of nuisance parameter.- 8.4. Asymptotic sufficiency and ancillarity.- 8.5. Reconstruction of estimator from those of independent samples.- 8.6. Notes.- References.- Subject Indices.