Statistical inference based on divergence measures

Statistical Inference Based on Divergence Measures......Page 4
Preface......Page 7
Contents......Page 12
References......Page 0
1.1. Introduction......Page 18
1.2. Phi-divergence Measures between Two Probability Distributions: Definition and Properties......Page 20
1.2.1. Basic Properties of the Phi-divergence Measures......Page 25
1.3. Other Divergence Measures between Two Probability Distributions......Page 35
1.3.1. Entropy Measures......Page 36
1.3.2. Burbea and Rao Divergence Measures......Page 42
1.4. Divergence among k Populations......Page 44
1.5. Phi-disparities......Page 46
1.6. Exercises......Page 48
1.7. Answers to Exercises......Page 51
2.1. Introduction......Page 71
2.2. Phi-entropies. Asymptotic Distribution......Page 74
2.3.1. Test for a Predicted Value of the Entropy of a Population (Diversity of a Population)......Page 80
2.3.3. Test for the Equality of the Entropies of r Independent Populations......Page 81
An Application to Equality of Variances in Normal Populations......Page 86
2.3.5. Confidence Intervals......Page 90
2.4. Multinomial Populations: Asymptotic Distributions......Page 91
2.4.1. Test of Discrete Uniformity......Page 100
2.5. Maximum Entropy Principle and Statistical Inference on Condensed Ordered Data......Page 103
2.6. Exercises......Page 109
2.7. Answers to Exercises......Page 114
3.1. Introduction......Page 129
3.2. Phi-divergences and Goodness-of-fit with Fixed Number of Classes......Page 132
3.3. Phi-divergence Test Statistics under Sparseness Assumptions......Page 141
3.4.1. Goodness-of-fit with Quantile Characterization......Page 148
Stationary Distribution......Page 151
Chain Markov and Order......Page 155
3.4.3. Misclassified Data......Page 156
3.4.4. Goodness-of-fit for and against Order Restrictions......Page 160
3.5. Exercises......Page 162
3.6. Answers to Exercises......Page 166
4.1. Introduction......Page 180
4.2.1. Pitman Asymptotic Relative Efficiency......Page 181
4.2.2. Bahadur Efficiency......Page 183
4.2.3. Approximations to the Power Function: Comparisons......Page 189
4.3. Exact and Asymptotic Moments: Comparison......Page 190
4.3.1. Under the Null Hypothesis......Page 191
4.3.2. Under Contiguous Alternative Hypotheses......Page 200
4.3.3. Corrected Phi-divergence Test Statistic......Page 204
4.4. A Second Order Approximation to the Exact Distribution......Page 205
4.5. Exact Powers Based on Exact Critical Regions......Page 209
4.6. Small Sample Comparisons for the Phi-divergence Test Statistics......Page 213
4.7. Exercises......Page 218
4.8. Answers to Exercises......Page 219
5.1. Introduction......Page 228
5.2. Maximum Likelihood and Minimum Phi-divergence Estimators......Page 230
5.2.1. Minimum Power-divergence Estimators in Normal and Weibull Populations......Page 235
5.3. Properties of the Minimum Phi-divergence Estimator......Page 239
5.3.1. Asymptotic Properties......Page 242
5.3.2. Minimum Phi-divergence Functional Robustness......Page 248
5.4. Normal Mixtures: Minimum Phi-divergence Estimator......Page 250
5.5. Minimum Phi-divergence Estimator with Constraints: Properties......Page 259
5.6. Exercises......Page 262
5.7. Answers to Exercises......Page 264
6.1. Introduction......Page 272
6.2. Asymptotic Distribution with Fixed Number of Classes......Page 274
6.3.1. Maximum Likelihood Estimator Based on Original Data and Test Statistics Based on Phi-divergences......Page 286
6.3.2. Goodness-of-fit with Quantile Characterization......Page 291
6.3.3. Estimation from an Independent Sample......Page 293
6.3.4. Goodness-of-fit with Dependent Observations......Page 294
6.3.5. Goodness-of-fit with Constraints......Page 296
6.4. Exercises......Page 297
6.5. Answers to Exercises......Page 299
7.1. Introduction......Page 312
7.2. Loglinear Models: Definition......Page 320
7.3. Asymptotic Results for Minimum Phi-divergence Estimators in Loglinear Models......Page 322
7.4. Testing in Loglinear Models......Page 324
7.5. Simulation Study......Page 340
7.6. Exercises......Page 351
7.7. Answers to Exercises......Page 355
8.1. Introduction......Page 365
8.2. Independence......Page 366
8.2.1. Restricted Minimum Phi-divergence Estimator......Page 368
8.2.2. Test of Independence......Page 374
Power of the Test......Page 376
8.3. Symmetry......Page 382
8.3.1. Test of Symmetry......Page 387
Power of the Test......Page 389
8.3.2. Symmetry in a Three-way Contingence Table......Page 392
8.4. Marginal Homogeneity......Page 396
8.5. Quasi-symmetry......Page 402
8.6. Homogeneity......Page 408
8.7. Exercises......Page 413
8.8. Answers to Exercises......Page 414
9.1. Introduction......Page 421
9.2. Simple Null Hypotheses: Wald, Rao, Wilks and Phi-divergence Test Statistics......Page 422
9.3. Composite Null Hypothesis......Page 434
9.4. Multi-sample Problem......Page 444
9.5. Some Topics in Multivariate Analysis......Page 450
9.6. Exercises......Page 451
9.7. Answers to Exercises......Page 454
References......Page 473