Statistical inference : the minimum distance approach.

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Basu, A., H. Shioya, et al. (2011). Statistical inference : the minimum distance approach.
Contents
Preface xv
Acknowledgments xix
1 Introduction 1
1.1 General Notation . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Some Background and Relevant Definitions . . . . . . . . . . 7
1.3.1 Fisher Information . . . . . . . . . . . . . . . . . . . . 7
1.3.2 First-Order Efficiency . . . . . . . . . . . . . . . . . . 9
1.3.3 Second-Order Efficiency . . . . . . . . . . . . . . . . . 9
1.4 Parametric Inference Based on the Maximum
Likelihood Method . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Hypothesis Testing by Likelihood Methods . . . . . . 11
1.5 Statistical Functionals and Influence Function . . . . . . . . 14
1.6 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . 18
2 Statistical Distances 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Distances Based on Distribution Functions . . . . . . . . . . 22
2.3 Density-Based Distances . . . . . . . . . . . . . . . . . . . . 25
2.3.1 The Distances in Discrete Models . . . . . . . . . . . . 26
2.3.2 More on the Hellinger Distance . . . . . . . . . . . . . 33
2.3.3 The Minimum Distance Estimator and the Estimating
Equations . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.4 The Estimation Curvature . . . . . . . . . . . . . . . . 38
2.3.5 Controlling of Inliers . . . . . . . . . . . . . . . . . . . 39
2.3.6 The Robustified Likelihood Disparity . . . . . . . . . . 40
2.3.7 The Influence Function of the Minimum Distance
Estimators . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.8 φ-Divergences . . . . . . . . . . . . . . . . . . . . . . . 45
2.4 Minimum Hellinger Distance Estimation: Discrete Models . . 46
2.4.1 Consistency of the Minimum Hellinger Distance
Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.2 Asymptotic Normality of the Minimum Hellinger
Distance Estimator . . . . . . . . . . . . . . . . . . . . 52
ix
x
2.5 Minimum Distance Estimation Based on Disparities: Discrete
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Continuous Models 73
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Minimum Hellinger Distance Estimation . . . . . . . . . . . 75
3.2.1 The Minimum Hellinger Distance Functional . . . . . 75
3.2.2 The Asymptotic Distribution of the Minimum Hellinger
Distance Estimator . . . . . . . . . . . . . . . . . . . . 78
3.3 Estimation of Multivariate Location and Covariance . . . . . 83
3.4 A General Structure . . . . . . . . . . . . . . . . . . . . . . . 87
3.4.1 Disparities in This Class . . . . . . . . . . . . . . . . . 93
3.5 The Basu–Lindsay Approach for Continuous Data . . . . . . 94
3.5.1 Transparent Kernels . . . . . . . . . . . . . . . . . . . 98
3.5.2 The Influence Function of the Minimum Distance
Estimators for the Basu–Lindsay Approach . . . . . . 100
3.5.3 The Asymptotic Distribution of the Minimum
Distance Estimators . . . . . . . . . . . . . . . . . . . 102
3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4 Measures of Robustness and Computational Issues 115
4.1 The Residual Adjustment Function . . . . . . . . . . . . . . 116
4.2 The Graphical Interpretation of Robustness . . . . . . . . . . 118
4.3 The Generalized Hellinger Distance . . . . . . . . . . . . . . 126
4.3.1 Connection with Other Distances . . . . . . . . . . . . 129
4.4 Higher Order Influence Analysis . . . . . . . . . . . . . . . . 129
4.5 Higher Order Influence Analysis: Continuous
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.6 Asymptotic Breakdown Properties . . . . . . . . . . . . . . . 137
4.6.1 Breakdown Point of the Minimum Hellinger Distance
Estimator . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.6.2 The Breakdown Point for the Power Divergence Family 139
4.6.3 A General Form of the Breakdown Point . . . . . . . . 141
4.6.4 Breakdown Point for Multivariate Location and
Covariance Estimation . . . . . . . . . . . . . . . . . . 144
4.7 The α α α-Influence Function . . . . . . . . . . . . . . . . . . . . 147
4.8 Outlier Stability of Minimum Distance Estimators . . . . . . 149
4.8.1 Outlier Stability of the Estimating Functions . . . . . 152
4.8.2 Robustness of the Estimator . . . . . . . . . . . . . . 153
4.9 Contamination Envelopes . . . . . . . . . . . . . . . . . . . . 156
4.10 The Iteratively Reweighted Least Squares (IRLS) . . . . . . 160
4.10.1 Development of the Algorithm . . . . . . . . . . . . . 160
4.10.2 The Standard IREE . . . . . . . . . . . . . . . . . . . 163
4.10.3 Optimally Weighted IREE . . . . . . . . . . . . . . . . 164
xi
4.10.4 Step by Step Implementation . . . . . . . . . . . . . . 166
5 The Hypothesis Testing Problem 167
5.1 Disparity Difference Test: Hellinger Distance Case . . . . . . 167
5.2 Disparity Difference Tests in Discrete Models . . . . . . . . . 172
5.2.1 Second-Order Effects in Testing . . . . . . . . . . . . . 175
5.3 Disparity Difference Tests: The Continuous Case . . . . . . . 180
5.3.1 The Smoothed Model Approach . . . . . . . . . . . . 182
5.4 Power Breakdown of Disparity Difference Tests . . . . . . . . 184
5.5 Outlier Stability of Disparity Difference Tests . . . . . . . . . 186
5.5.1 The GHD and the Chi-Square Inflation Factor . . . . 189
5.6 The Two-Sample Problem . . . . . . . . . . . . . . . . . . . 191
6 Techniques for Inlier Modification 195
6.1 Minimum Distance Estimation: Inlier Correction in Small
Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.2 Penalized Distances . . . . . . . . . . . . . . . . . . . . . . . 197
6.2.1 The Penalized Hellinger Distance . . . . . . . . . . . . 198
6.2.2 Minimum Penalized Distance Estimators . . . . . . . 200
6.2.3 Asymptotic Distribution of the Minimum Penalized
Distance Estimator . . . . . . . . . . . . . . . . . . . . 201
6.2.4 Penalized Disparity Difference Tests: Asymptotic
Results . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.2.5 The Power Divergence Family versus the Blended
Weight Hellinger Distance Family . . . . . . . . . . . . 207
6.3 Combined Distances . . . . . . . . . . . . . . . . . . . . . . . 212
6.3.1 Asymptotic Distribution of the Minimum Combined
Distance Estimators . . . . . . . . . . . . . . . . . . . 216
6.4 ǫ-Combined Distances . . . . . . . . . . . . . . . . . . . . . . 222
6.5 Coupled Distances . . . . . . . . . . . . . . . . . . . . . . . . 225
6.6 The Inlier-Shrunk Distances . . . . . . . . . . . . . . . . . . 227
6.7 Numerical Simulations and Examples . . . . . . . . . . . . . 230
7 Weighted Likelihood Estimation 235
7.1 The Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . 236
7.1.1 The Disparity Weights . . . . . . . . . . . . . . . . . . 237
7.1.2 Influence Function and Standard Error . . . . . . . . . 242
7.1.3 The Mean Downweighting Parameter . . . . . . . . . 244
7.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.2 The Continuous Case . . . . . . . . . . . . . . . . . . . . . . 249
7.2.1 Influence Function and Standard Error: Continuous
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.2.2 The Mean Downweighting Parameter . . . . . . . . . 252
7.2.3 A Bootstrap Root Search . . . . . . . . . . . . . . . . 253
7.2.4 Asymptotic Results . . . . . . . . . . . . . . . . . . . 254
xii
7.2.5 Robustness of Estimating Equations . . . . . . . . . . 255
7.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.4 Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . 261
7.5 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 263
8 Multinomial Goodness-of-Fit Testing 265
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8.1.1 Chi-Square Goodness-of-Fit Tests . . . . . . . . . . . . 266
8.2 Asymptotic Distribution of the Goodness-of-Fit Statistics . . 267
8.2.1 The Disparity Statistics . . . . . . . . . . . . . . . . . 268
8.2.2 The Simple Null Hypothesis . . . . . . . . . . . . . . . 268
8.2.3 The Composite Null Hypothesis . . . . . . . . . . . . 270
8.2.4 Minimum Distance Inference versus Multinomial
Goodness-of-Fit . . . . . . . . . . . . . . . . . . . . . . 272
8.3 Exact Power Comparisons in Small Samples . . . . . . . . . 273
8.4 Choosing a Disparity to Minimize the Correction Terms . . . 277
8.5 Small Sample Comparisons of the Test Statistics . . . . . . . 280
8.5.1 The Power Divergence Family . . . . . . . . . . . . . . 280
8.5.2 The BWHD Family . . . . . . . . . . . . . . . . . . . 282
8.5.3 The BWCS Family . . . . . . . . . . . . . . . . . . . . 283
8.5.4 Derivation of F S (y) for a General Disparity Statistic . 284
8.6 Inlier Modified Statistics . . . . . . . . . . . . . . . . . . . . 286
8.6.1 The Penalized Disparity Statistics . . . . . . . . . . . 287
8.6.2 The Combined Disparity Statistics . . . . . . . . . . . 288
8.6.3 Numerical Studies . . . . . . . . . . . . . . . . . . . . 290
8.7 An Application: Kappa Statistics . . . . . . . . . . . . . . . 294
9 The Density Power Divergence 297
9.1 The Minimum L 2 Distance Estimator . . . . . . . . . . . . . 298
9.2 The Minimum Density Power Divergence Estimator . . . . . 300
9.2.1 Asymptotic Properties . . . . . . . . . . . . . . . . . . 303
9.2.2 Influence Function and Standard Error . . . . . . . . . 308
9.2.3 Special Parametric Families . . . . . . . . . . . . . . . 309
9.3 A Related Divergence Measure . . . . . . . . . . . . . . . . . 311
9.3.1 The JHHB Divergence . . . . . . . . . . . . . . . . . . 311
9.3.2 Formulae for Variances . . . . . . . . . . . . . . . . . . 314
9.3.3 Numerical Comparisons of the Two Methods . . . . . 316
9.3.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . 316
9.4 The Censored Survival Data Problem . . . . . . . . . . . . . 317
9.4.1 A Real Data Example . . . . . . . . . . . . . . . . . . 318
9.5 The Normal Mixture Model Problem . . . . . . . . . . . . . 322
9.6 Selection of Tuning Parameters . . . . . . . . . . . . . . . . . 323
9.7 Other Applications of the Density Power Divergence . . . . . 324
xiii
10 Other Applications 327
10.1 Censored Data . . . . . . . . . . . . . . . . . . . . . . . . . . 327
10.1.1 Minimum Hellinger Distance Estimation in the
Random Censorship Model . . . . . . . . . . . . . . . 327
10.1.2 Minimum Hellinger Distance Estimation Based on
Hazard Functions . . . . . . . . . . . . . . . . . . . . . 329
10.1.3 Power Divergence Statistics for Grouped Survival Data 330
10.2 Minimum Hellinger Distance Methods in Mixture Models . . 331
10.3 Minimum Distance Estimation Based on Grouped Data . . . 332
10.4 Semiparametric Problems . . . . . . . . . . . . . . . . . . . . 335
10.4.1 Two-Component Mixture Model . . . . . . . . . . . . 335
10.4.2 Two-Sample Semiparametric Model . . . . . . . . . . 336
10.5 Other Miscellaneous Topics . . . . . . . . . . . . . . . . . . . 337
11 Distance Measures in Information and Engineering 339
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
11.2 Entropies and Divergences . . . . . . . . . . . . . . . . . . . 340
11.3 Csiszár’s f -Divergence . . . . . . . . . . . . . . . . . . . . . . 341
11.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 341
11.3.2 Range of the f -Divergence . . . . . . . . . . . . . . . . 343
11.3.3 Inequalities Involving f -Divergences . . . . . . . . . . 345
11.3.4 Other Related Results . . . . . . . . . . . . . . . . . . 346
11.4 The Bregman Divergence . . . . . . . . . . . . . . . . . . . . 346
11.5 Extended f -Divergences . . . . . . . . . . . . . . . . . . . . . 347
11.5.1 f -Divergences for Nonnegative Functions . . . . . . . . 347
11.5.2 Another Extension of the f -Divergence . . . . . . . . . 351
11.6 Additional Remarks . . . . . . . . . . . . . . . . . . . . . . . 352
12 Applications to Other Models 353
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.2 Preliminaries for Other Models . . . . . . . . . . . . . . . . . 354
12.3 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . 356
12.3.1 Models and Previous Works . . . . . . . . . . . . . . . 356
12.3.2 Feed-Forward Neural Networks . . . . . . . . . . . . . 356
12.3.3 Training Feed-Forward Neural Networks . . . . . . . . 357
12.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . 360
12.3.5 Related Works . . . . . . . . . . . . . . . . . . . . . . 360
12.4 Fuzzy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 361
12.4.1 Fundamental Elements of Fuzzy Sets . . . . . . . . . . 361
12.4.2 Measures of Fuzzy Sets . . . . . . . . . . . . . . . . . 362
12.4.3 Generalized Fuzzy Divergence . . . . . . . . . . . . . . 364
12.5 Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . 365
12.5.1 Diffractive Imaging . . . . . . . . . . . . . . . . . . . . 365
12.5.2 Algorithms for Phase Retrieval . . . . . . . . . . . . . 367
12.5.3 Statistical-Distance-Based Phase Retrieval Algorithm 368
xiv
12.5.4 Numerical Example . . . . . . . . . . . . . . . . . . . 369
12.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Bibliography 373

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关键词：Statistical statistica statistic Inference Distance distance minimum General 2011

太贵了吧...

您好，这个minimum distance 的书正好是我需要的，但是买不起，可以发我邮箱吗？1030520198@qq.com.谢谢!

还是自己买了（论坛币太少了真不禁花），不过书质量不错~

lena7 发表于 2014-11-20 19:50
还是自己买了（论坛币太少了真不禁花），不过书质量不错~

对不起，刚刚看到你的留言，若有其他需要的书，再给我留言（但我不能每天登陆，有时需要等一段时间）