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Title: Limit Theorems in Change-Point Analysis
Authors: M. Csorgo L. Horvath
Foreword xi
Preface xiii
Chapter 1. The Likelihood Approach 1
1.1 Processes Based on the Likelihood Ratio 1
1.2 Proofs for Approximations of Log Likelihood Ratio
Processes 7
1.3 Asymptotics for the Maximum Log Likelihood Ratio
Test 21
1.4 Some Examples 27
1.5 The Maximal Log Likelihood Ratio Test under
Alternatives 34
1.6 Estimators for the Time of Change 50
1.7 More Applications of the Likelihood Principle 57
1.7.1 Fisher-Score Change Processes 57
1.7.2 Maximally Selected Chi-Square Statistics 58
1.7.3 Multiple Changes 60
1.7.4 The Epidemic Alternative 62
1.7.5 Estimating the Number of Changes 64
1.8 Further Results and Remarks 65
Chapter 2. Nonparametric Methods 67
2.1 Change in the Mean 67
2.2 Change in Location 89
2.3 Wilcoxon-Type Statistics 98
2.4 Processes Based on U-Statistics 110
2.5 Pontograms 135
2.6 Applications of Empirical Distributions 152
2.7 Sequential Ranks 157
2.8 More Nonparametric Methods 170
2.8.1 Estimators for the Time of Change 170
2.8.2 Tests Based on M-Estimators 178
2.8.3 Moving Sums 179
2.8.4 Nonparametric Tests for Epidemic Alternatives 181
2.8.5 Smooth Changes 186
2.8.6 Multiple Change-Points 187
2.8.7 Change in the Variance 188
2.8.8 Detection of Changes in Angular Data 190
2.9 Further Results and Remarks 194
Chapter 3 Linear Models 199
3.1 On the Maximum Likelihood Ratio Method in Regression 199
3.1.1 Change in Regression Coefficients 199
3.1.2Change in Regression Coefficients and Variance 203
3.1.3 Proofs of Theorems 3.1.1 and 3.1.2 206
3.1.4 Proofs of Theorems 3.1.3 and 3.1.4 219
3.1.5 The Union-Intersection Test 224
3.2 Tests Based on Comparison of Estimators 230
3.2.1 Direct Tests for Trend 231
3.2.2 Chow's Tests 247
3.3 Sums of Regression Residuals 248
3.4 Sums of Recursive Regression Residuals 261
3.5 Consistency of Tests in Sections 3.1-3.4 268
3.5.1 The Likelihood Ratio Test for Change in
Regression Coefficients 268
3.5.2 The Likelihood Ratio Test for Change in the
Regression Coefficients and the Variance 274
3.5.3 Consistency of the Union-Intersection Test 276
3.5.4 Consistency of Direct Tests 279
3.5.5 Maximum of Sums of Regression 280
Residuals
3.6 Estimators for the Time of Change 281
3.7 Further Results and Remarks 297
Chapter 4. Dependent Observations 299
4.1 Weakly Dependent Sequences 299
4.2 Procedures Based on Residuals 312
4.2.1 Stationary ARMA(p,<?) Processes 312
4.2.2 Explosive AR{p) processes 321
4.2.3 Random Walk Model 327
4.3 Change in the Mean of Strongly Dependent
Sequences 329
4.4 More on Dependent Observations 354
4.4.1 Back to Maximal Likelihood Ratio 354
4.4.2 Spectral Empirical Distributions 356
4.5 Further Results and Remarks 358
Appendix
350
A.I Approximations for Partial Sums of Random Variables and Random Vectors
359
A.2 Approximations for Empirical and Quantn'e Processes
361
A.3 Limit Theorems for Ornstein-Uhlenbeck and Related Processes
363
A.4 Limit Theorems for Standardized Partial Sums and Related Processes
368
A.5 Integral Tests for some Gaussian Processes
372
A.6 Contiguity
374
A.7 Integral Tests for Functionals of Kiefer Processes
378
References
387
Author Index
405
Subject Index
409
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