Geometrical Foundations of Asymptotic Inference

Geometrical Foundations of Asymptotic Inference......Page 5
Contents......Page 7
Preface......Page 13
1 Overview and Preliminaries......Page 15
1.1.1 Part I......Page 16
1.1.2 Part II......Page 17
1.1.3 Part III......Page 18
1.2 Notation......Page 20
1.2.2 Differentiation......Page 21
1.2.3 Tensor Notation......Page 23
1.2.4 Connection Notation......Page 24
PART I ONE-PARAMETER CURVED EXPONENTIAL FAMILIES......Page 25
2.1 Introduction......Page 27
2.2.1 Basic Properties......Page 28
2.2.2 Asymptotics......Page 33
2.3 Curved Exponential Families: Definition and Examples......Page 41
2.3.1 Definition and Basic Properties......Page 43
2.3.2 Examples......Page 47
2.4.1 Estimating Equations......Page 53
2.4.2 Auxiliary Spaces......Page 55
2.5.1 Information and Sufficiency......Page 57
2.5.2 The Information Inner Product......Page 60
2.5.3 Observed Information......Page 63
2.5.4 The Kullback–Leibler Divergence......Page 65
2.6.1 Consistency and Asymptotic Normality......Page 67
2.6.2 Efficiency......Page 70
2.7 Bibliographical Remarks......Page 74
3.1 Introduction......Page 75
3.2.1 Definition and Calculation......Page 76
3.2.2 Examples......Page 78
3.3 Information Loss and Local Sufficiency......Page 80
3.3.1 Information Loss......Page 81
3.3.2 Information Recovery......Page 87
3.3.3 Local Sufficiency......Page 89
3.4.2 Deficiency......Page 92
3.4.4 The Fisher Scoring Algorithm......Page 93
3.5 Edgeworth Expansions......Page 94
3.6 Posterior Expansions......Page 96
3.7 Extensions......Page 103
3.7.1 Efron\'s General Formula......Page 104
3.7.2 Small-Dispersion Asymptotics......Page 106
3.8 Bibliographical Remarks......Page 107
PART II MULTIPARAMETER CURVED EXPONENTIAL FAMILIES......Page 109
4.1 Introduction......Page 111
4.2 Multiparameter Curved Exponential Families......Page 112
4.3.1 Curvature and Information Loss......Page 116
4.3.2 Asymptotic Risk and Bias......Page 117
4.3.3 Interpretation in Nonlinear Regression......Page 118
4.3.4 Statistical Curvature in General Families......Page 119
4.4 Information Loss and Sufficiency......Page 120
4.5 Multivariate Edgeworth Series......Page 123
4.6 Posterior Expansions......Page 125
4.7 Bibliographical Remarks......Page 126
5.1 Introduction......Page 127
5.2.1 Normal Regression Model......Page 128
5.2.3 Tangent Bundle......Page 130
5.3.1 Preliminary Concepts......Page 134
5.3.2 A Vector Space Structure......Page 137
5.3.3 The Fisher Information Inner Product......Page 139
5.3.4 Estimation Algorithms......Page 141
5.4 Measures of Influence......Page 144
5.4.1 Normal Linear Regression......Page 145
5.4.2 Exponential Family Regression......Page 146
5.5 Sensitivity Analysis of the Moment Structure......Page 148
5.5.1 Quasi-Likelihood Functions......Page 149
5.5.2 The Measures DL and LDLa......Page 150
5.5.3 Perturbations of the Moment Structure......Page 153
5.6 Bibliographical Remarks......Page 157
6.1 Introduction......Page 158
6.2 Background on Nonlinear Regression......Page 162
6.2.1 Asymptotic Normality......Page 163
6.2.2 Curvature Measures of Nonlinearity......Page 168
6.3.1 Generalizing the Standardized Second-Derivative Array......Page 172
6.3.2 Curvature Measures......Page 174
6.4 Summaries of the Observed Third-Derivative Array......Page 176
6.5 Examples......Page 179
6.6 Bibliographical Remarks......Page 185
PART III DIFFERENTIAL-GEOMETRIC METHODS......Page 187
7.1 Introduction......Page 189
7.2.1 Spherical Representation......Page 190
7.2.2 Information Distance, Hellinger Distance, and Kullback–Leibler Number......Page 192
7.2.3 Jeffreys\' Prior......Page 193
7.2.4 Orthogonal Parameters......Page 194
7.2.5 Imbedded Subfamilies and Asymptotically Chi-Squared Statistics......Page 195
7.2.6 Inference Regions Based on Information Distance......Page 197
7.3 Manifolds of Densities......Page 198
7.4.1 Basic Construction and Properties......Page 199
7.4.2 Multinomial Geometry......Page 203
7.4.3 Spherical Normal Geometry......Page 205
7.4.4 Location-Scale Families......Page 206
7.4.5 General Multivariate Normal Families......Page 210
7.5 Hessians That Define Metrics......Page 211
7.6.1 Goodness-of-Fit Testing......Page 214
7.6.2 Reference Priors: Uniformity, Symmetry, and Jeffreys\' Rules......Page 224
7.6.3 Orthogonal Parameters......Page 226
7.7 Bibliographical Remarks......Page 227
8.2 Non-Riemannian Connections......Page 229
8.3 Statistical Manifolds......Page 230
8.4 Curvature and Supermanifolds......Page 232
8.5 Applications......Page 233
8.5.1 Information Recovery......Page 235
8.5.2 Conditional Inference......Page 239
8.5.3 Efficient Estimators......Page 241
8.5.4 Efficient Tests......Page 244
8.5.5 Parameterizations and α-Connections......Page 247
8.6 Bibliographical Remarks......Page 250
9.1 Introduction......Page 252
9.2 Minimum Divergence Estimation......Page 253
9.2.1 Divergence and Quasi-Likelihood......Page 254
9.2.2 Minimum Chi-Squared Estimation......Page 257
9.3 Divergence Geometry......Page 259
9.3.1 Riemannian Manifold......Page 260
9.3.2 Flat Statistical Manifold......Page 265
9.3.3 Law of Cosines......Page 268
9.4 Divergences Compatible with a Metric......Page 269
9.4.1 f-Divergence......Page 270
9.4.2 Auxiliary Submanifolds......Page 273
9.4.3 Power Family of Divergences......Page 276
9.4.4 I-Divergence Statistics......Page 280
9.4.5 α- and λ-Connections......Page 281
9.5.1 Asymptotic Efficiency......Page 282
9.5.2 Divergence Sufficiency......Page 285
9.5.3 Extended Quasi-Likelihood......Page 287
9.6 Bibliographical Remarks......Page 291
10.1 Introduction......Page 292
10.2.1 Exponential Bundle......Page 294
10.2.2 Mixture Bundle......Page 296
10.2.3 Other Finite Vector Bundles......Page 297
10.3 Hilbert Bundle......Page 298
10.3.1 Construction of M × B......Page 299
10.3.2 Connections and Parallel Transports......Page 300
10.3.3 Estimating Functions......Page 302
10.4 Orthogeodesic Models......Page 305
10.5 Statistical Fiber Bundles......Page 307
10.6 Preferred Point Geometries......Page 309
10.7 Bibliographical Remarks......Page 313
Appendix A Diffeomorphisms and the Inverse Function Theorem......Page 314
B.1 Basic Results......Page 317
B.2 Reciprocal of the Radius of the Best-Fitting Circle......Page 320
B.3 Rate of Change of Direction......Page 321
C.1 Manifolds......Page 323
C.3 The Tangent Space......Page 326
C.4 Connections......Page 329
C.4.1 Geodesics......Page 331
C.4.2 Exponential Maps......Page 332
C.5 Riemannian Metric......Page 333
C.5.1 Geodesics and the Metric Connection......Page 334
C.5.2 Submanifolds......Page 336
C.6.2 Comments on Tensor Notation......Page 337
C.8 The Measure Determined by a Metric......Page 342
C.9 Curvature......Page 344
C.9.1 Gaussian and Riemannian Curvature......Page 345
C.9.2 Submanifolds and Imbedding Curvature......Page 348
Appendix D A Coordinate-Free Definition of Weak Sphericity......Page 351
References......Page 354
Symbol Index......Page 363
Index......Page 365