Chi-squared Goodness-of-fit Tests for Censored Data

This book is devoted to the problems of construction and application of chi-squared goodness-of-fit tests for complete and censored data. Classical chi-squared tests assume that unknown distribution parameters are estimated using grouped data, but in practice this assumption is often forgotten. In this book, we consider modified chi-squared tests, which do not suffer from such a drawback. The authors provide examples of chi-squared tests for various distributions widely used in practice, and also consider chi-squared tests for the parametric proportional hazards model and accelerated failure time model, which are widely used in reliability and survival analysis. Particular attention is paid to the choice of grouping intervals and simulations.

This book covers recent innovations in the field as well as important results previously only published in Russian. Chi-squared tests are compared with other goodness-of-fit tests (such as the Cramer-von Mises-Smirnov, Anderson-Darling and Zhang tests) in terms of power when testing close competing hypotheses.

Introduction ix

Chapter 1. Chi-squared Goodness-of-fit Tests for Complete Data 1

1.1. Classical Pearson’s chi-squared test 1

1.2. Joint distribution of Xn(θ∗n)and√n(θ∗n−θ) 3

1.3. Parameter estimation based on complete data Lemma of Chernoff and Lehmann 5

1.4. Parameter estimation based on grouped data. Theorem of Fisher 10

1.5. Nikulin-Rao-Robson chi-squared test 12

1.6. Other modifications 18

1.7. The choice of grouping intervals 20

Chapter 2. Chi-squared Test for Censored Data 31

2.1. Generalized Pearson-Fisher chi-squared test 32

2.2. Maximum likelihood estimators for censored data 34

2.3. Nikulin-Rao-Robson chi-squared test for censored data 38

2.4. The choice of grouping intervals 45

2.4.1. Equifrequent grouping (EFG) 45

2.4.2. Intervals with equal expected numbers of failures (EENFG) 46

2.4.3. Optimal grouping (OptG) 48

2.5. Chi-squared tests for specific families of distributions 51

2.5.1. Exponential distribution 51

2.5.2. Weibull distribution 55

2.5.3. Lognormal distribution 60

2.5.4. Loglogistic distribution 63

2.5.5. Gompertz distribution 67

Chapter 3. Comparison of the Chi-squared Goodness-of-fit Test with Other Tests 71

3.1. Tests based on the difference between non-parametric and parametric estimators 71

3.2. Comparison of goodness-of-fit tests for complete data 76

3.3. Comparison of goodness-of-fit tests for censored data 79

3.3.1. Lognormal-generalized Weibull pair of competing hypotheses 80

3.3.2. Exponential-Weibull pair of competing hypotheses 82

3.3.3. Weibull-generalized Weibull pairs of competing hypotheses 84

Chapter 4. Chi-squared Goodness-of-fit Tests for Regression Models 87

4.1. Data and the idea of chi-squared test construction 89

4.2. Asymptotic distribution of the random vector Z 91

4.3. Test statistic 96

4.4. Choice of random grouping intervals 97

4.4.1. Test for the exponential AFT model 99

4.4.2. Tests for the scale-shape AFT models with constant covariates 101

4.4.3. Test for the Weibull AFT model with step-stresses 108

Appendices 111

Appendix 1 113

Appendix 2 125

Bibliography 131

Index 141

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