Abdolreza, Sayyareh (2007) Test of fit and model selection based on likelihood function. PhD thesis Statistique, ISPED, Unité 875 Biostatistique, Université de Bordeaux 2, AgroParistech 2007AGPT0020 p.205.
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Abstract
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.
| Item Type: | PhD Thesis (PhD) |
|---|---|
| Thesis Supervisor: | Bar-hen, Avner |
| Date: | 22 June 2007 |
| Board of examiners: | Biernacki, Christophe and Saracco, Jérôme and Daudin, Jean Jacques and Commenges, Daniel |
| Ecole Doctorale: | ED 435 AGRICULTURE, ALIMENTATION, BIOLOGIE, ENVIRONNEMENTS ET SANTE |
| Discipline: | Statistique |
| Collection (Fonds): | AgroParistech |
| Institution: | AgroParistech |
| Department: | ISPED, Unité 875 Biostatistique, Université de Bordeaux 2 |
| Subjects: | 1. Mathematics and Applications |
| Uncontrolled Keywords: | Akaike criterion, Confidence interval, Kullback leibler, Logistic regression, Model selection, Multiple regression, Variable selection |
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Table of content
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
16
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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| ID Code: | 3400 |
|---|---|
| Deposited By: | Nadine Pontal |
| Deposited On: | 12 February 2008 |
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