Differential Geometrical Methods in Statistics (Lecture notes in statistics)
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Number Of Pages:   299
Publication Date:   1990-01
Sales Rank:   4938579
ISBN / ASIN:   3540960562
EAN:   9783540960560
Binding:   Paperback
Manufacturer:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Studio:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K


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
aapter 1. Introduction
Ш? I. GEOMETRICAL STRUCTURES OF A FAMILY OF PROBABILITY
DISTRIBUTIONS 11
tiapter 2. Differential Geometry of Statistical Models 11
2.1. Manifold of statistical model 11
2.2. Tangent space 16
2.3. Riemannian metric and Fisher information 25
2.4. Affine connection 32
2.5. Statistical α-connection 38
2.6. Curvature and torsion 43
2.7. Imbedding and submanifold 49
2.8. Family of ancillary submanifolds 54
2.9. Notes 63
hapter 3. α-Divergence and α-Projection in Statistical Manifold 66
3.1. α-representation 66
3.2. Dual affine connections 70
3.3. α-family of distributions 73
3.4. Duality in α-flat manifolds 79
3.5. α-divergence 84
3.6. α-projection 89
3.7. On geometry of function space of distributions 93
3.8. Remarks on possible divergence, metric and
connection in statistical manifold 96
3.9.. Notes 102
PART П. HIGHER-ORDER ASYMPTOTIC THEORY OF STATISTICAL INFERENCE
IN CURVED EXPONENTIAL FAMILIES 104
Chapter 4. Curved Exponential Families and Edgeworth Expansions 104
4.1. Exponential family 104
4.2. Curved exponential family 108
4.3. Geometrical aspects of statistical inference 115
4.4. Edgeworth expansion 120
4.5. Notes 127
Chapter 5. Asymptotic Theory of Estimation 128
5.1. Consistency and efficiency of estimators 128
5.2. Second- and third-order efficient estimator 131
5.3. Third-order error of estimator without bias
correction 141
5.4. Ancillary family depending on the number of
observations 145
5.5. Effects of parametrization 148
5.6. Geometrical aspects of jacknifing 156
5.7. Notes 159
Chapter 6. Asymptotic Theory of Tests and Interval Estimators 161
6.1. Ancillary family associated with a test 161
6.2. Asymptotic evaluations of tests: scalar parameter
case 171
6.3. Characteristics of widely used efficient tests:
Scalar parameter case 181
6.4. Conditional test 190
6.5. Asymptotic properties of interval estimators 193
6.6. Asymptotic evaluations of tests: general case 197
6.6. Notes 208
napter 7. Information, Ancillarity and Conditional Inference 210
7.1. Conditional information, asymptotic sufficiency
and asymptotic ancillarity 210
7.2. Conditionl inference 217
7.3. Pooling independent observations 231
7.4. Complete decomposition of information 236
7.5. Notes 241
tiapter 8. Statistical Inference in the Presence of Nuisance
Parameters 244
8.1. Orthogonal parametrization and orthogonalized
information 244
8.2. Higher-order efficiency of estimators 255
8.3. The amount of information carried by knowledge
of nuisance parameter 257
8.4. Asymptotic sufficiency and ancillarity 261
8.5. Reconstruction of estimator from those of
independent samples 268
8.6. Notes 273
EFERENCES 276
UBJECT INDICES
291