Test of fit and model selection based on likelihood function.

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Abdolreza, Sayyareh (2007) Test of fit and model selection based on likelihood function. Doctorat Statistique, ISPED, Unité 875 Biostatistique, Université de Bordeaux 2, AgroParistech 2007AGPT0020 p.205.

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Résumé

Notre travail port sur l’inf´erence au sujet de l’AIC (un cas de vraisemblance

p`enalis´ee) d’Akaike (1973), o`u comme estimateur de divergence de Kullback-Leibler est

intimement reli´ee `a l’estimateur de maximum de vraisemblance. Comme une partie de la statistique

inf´erentielle, dans le contexte de test d’hypoth`ese, la divergence de Kullback-Leibler et le lemme

de Neyman-Pearson sont deux concepts fondamentaux. Tous les deux sont au sujet du rapports de

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vraisemblance. Neyman-Pearson est au sujet du taux d’erreur du test du rapport de vraisemblance

et la divergence de Kullback-Leibler est l’esp´erance du rapport de log-vraisemblance.

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Table des Matières

1 Introduction 19

1.1 Our Objective - 27

1.2 Plan of Thesis - 29

2 Reminders about models

and some asymptotic results 30

2.1 Models - 30

2.2 Model Selection - 32

2.3 Goal of Model Selection and its means - 34

2.4 Nested and Non-Nested Models - 35

2.5 Probability Metrics - 36

2.6 Akaike framework and his Theorem - 37

2.7 Complexity in model selection - 38

2.8 Asymptotic theory - 42

2.9 Goodness of Fit Test and

Classical Hypothesis Testing - 43

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CONTENTS CONTENTS

2.10 Reminder on Theorems and Lemmas - 45

3 Reminder on Goodness of Fit Tests 47

3.1 Testing fit to a fixed distribution - 47

3.1.1 Basic Goodness of Fit Test - 48

3.1.2 Tests on the basis of Functional Distance - 49

3.2 Adaptation of tests coming from the fixed-distribution - 51

3.3 Tests on the basis of Correlation and Regression - 52

3.4 Tests on the basis of Likelihood Functions - 54

3.4.1 Berk-Jones’s statistics - 54

3.4.2 Generalized Linear Models (GLMs) and Deviance - 55

4 Motivation to Model Selection Tests 61

4.1 Introduction - 61

4.2 Assumptions - 63

4.3 Likelihood Function and

Maximum Likelihood Estimator - 65

4.3.1 Correctly Specified and Mis-Specified models - 68

4.4 Metrics on spaces of probability - 69

4.4.1 Kullback-Leibler Discrepancy (divergence) - 73

4.5 Consistency of Maximum Likelihood

Estimator - 76

4.6 Akaike Information Criterion (AIC) - 77

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CONTENTS CONTENTS

4.7 Distribution of Maximum Likelihood

Estimator - 79

5 Proposed test for Goodness of Fit Test:

A test based on empirical likelihood ratio 82

5.1 Introduction - 82

5.2 Our objective - 84

5.3 Union-Intersection Test - 85

5.4 Proposed test based on empirical

likelihood ratio - 86

5.4.1 Level of test - 89

5.4.2 Comparison with Berk-Jones’s test - 89

5.4.3 Bahadur efficiency of proposed test - 89

6 Proposed Model selection tests based on likelihood and AIC 92

6.1 Introduction - 92

6.2 Known parameters case - 98

6.3 Unknown parameters case - 106

6.4 Test function - 110

6.5 Variance estimation - 111

6.6 Distribution of Tn under H1

and Power of Test - 123

6.6.1 Distribution of Test Statistic Tn under H1 - 123

6.6.2 Power of Test - 127

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CONTENTS CONTENTS

6.7 Consistency of Test - 130

6.7.1 Power computation - 131

6.7.2 Invariance - 132

7 Test For Model Selection based on difference of AIC’s:

application to tracking interval for DEKL 136

7.1 Introduction - 136

7.2 Objective - 138

7.3 Non-Nested Models comparison - 140

7.3.1 Motivation to Confidence Interval construction - 142

7.3.2 Confidence Interval for DEKL - 144

7.4 Logistic Regression: - 149

8 Conclusion and perspective 156

9 Bibliography 161

10 Appendix 165

I APPENDIX A 166

10.1 Introduction - 1

10.2 Expected Kullback-Leibler Criteria and AIC - 5

10.3 Hypothesis Testing - 7

10.4 Simulation - 10

10.4.1 exploration of our result - 10

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CONTENTS CONTENTS

10.4.2 Application to The Multiple Regression Model - 11

II APPENDIX B 20

10.5 Introduction - 22

10.6 Theory about inference of differences of AIC criteria - 24

10.6.1 Estimating a difference of Kullback-Leibler divergences - 24

10.6.2 Tracking interval for a difference of Kullback-Leibler divergences - 27

10.6.3 Extension to regression models - 29

10.7 Application to logistic regression: a simulation study - 30

10.8 Choice of the best coding of age in a study of depression - 32

10.8.1 The Paquid study - 32

10.9 Discussion - 34

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