Lionel, Truquet (2008) Propriétés théoriques et applications en statistique et en simulation de processus et de champs aléatoires stationnaires. PhD thesis SCIENCES SPÉCIALITÉ MATHÉMATIQUES APPLIQUÉES, CREST Laboratoire de Statistiques , ENSAE p.198.
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Abstract
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| Item Type: | PhD Thesis (PhD) |
|---|---|
| PhD Supervisor: | Doukhan, Paul and Bardet, Jean-Marc |
| Date: | 10 December 2008 |
| Board of examiners: | Delyon, Bernard and Kokoszka, Piotr and Shao, Qi-Man and Davydov, Youri and Guyon, Xavier and Latour, Alain and Merlevede, Florence and Zakoian, Jean-Michel |
| Ecole Doctorale: | UNIVERSITÉ PARIS I PANTHÉON-SORBONNE |
| Discipline: | SCIENCES SPÉCIALITÉ MATHÉMATIQUES APPLIQUÉES |
| Collection (Fonds): | ENSAE ParisTech |
| Institution: | ENSAE |
| Department: | CREST Laboratoire de Statistiques |
| Subjects: | 1. Mathematics and Applications |
| Uncontrolled Keywords: | Statistiques – Processus stationnaires – Champs aléatoires – Dépendance faible, Statistics – Stationary process – Random fields – Weak dependence |
| ID Code: | 5120 |
| Deposited By: | Luc Langouet |
| Deposited On: | 13 May 2009 |
Table of content
1 Synthèse des travaux 1
1.1 Dépendance faible des champs aléatoires - 1
1.1.1 Champs aléatoires auto-régressifs - 3
1.1.2 La faible dépendance - 6
1.1.3 Le principe d'invariance fort - 7
1.2 Quelques problèmes d'estimation paramétriques et non paramétriques - 8
1.2.1 Simulation de textures par champs de Markov - 8
1.2.2 Un modèle de type bilinéaire pour des séries à valeurs entières - 13
1.2.3 QMLE lissé et estimation des modèles LARCH - 17
2 A xed point approach to model random elds 23
2.1 Introduction - 23
2.2 Main results - 25
2.2.1 Weak dependence - 25
2.2.2 Random elds with innite interactions - 25
2.2.3 Causality - 28
2.2.4 Simulation of the model - 30
2.3 Examples - 33
2.3.1 Finite interactions random elds - 33
2.3.2 Linear elds - 33
2.3.3 LARCH(1) random elds - 34
2.3.4 Non linear ARCH(1) random elds - 35
2.3.5 Mean eld type model - 35
2.4 Proofs - 36
2.4.1 Proof of lemma 2.1 - 36
2.4.2 Proof of the existence theorem 4.2 - 37
2.4.3 Proof of theorem 2.2 - 37
2.4.4 Proof of proposition 2.1 - 39
2.4.5 General case - 43
2.4.6 Results on causality - 43
2.4.7 Proof of theorem 2.3 - 45
2.4.8 Proof of lemma 2.6 - 46
2.4.9 Proofs for the section 2.3 - 46
3 Weak Dependence, Models and Some Applications 51
3.1 Introduction - 51
3.2 Weak dependence - 52
3.2.1 Independence - 52
3.2.2 Mixing - 52
3.2.3 Denition - 54
3.2.4 Basic examples - 56
3.2.5 Botanic of the models - 56
3.3 Least squares estimation of ARCH(1) processes - 58
3.3.1 Denition, identiability, existence and consistency - 59
3.3.2 Asymptotic normality - 59
3.3.3 Eective estimation procedures - 62
3.4 Random elds - 63
3.4.1 Moment inequalities for partial sums - 64
3.4.2 A central limit theorem for weakly dependent random elds - 69
3.4.3 Donsker invariance principle for random elds - 72
4 Strong invariance principle for a new class of weakly dependent random elds 77
4.1 Introduction - 77
4.2 Covariances estimates for Bernoulli shifts - 79
4.3 Moment inequalities for weakly dependent elds - 82
4.4 The strong invariance principle for spatial Bernoulli shifts - 84
4.5 Proof of Theorem 4:2 - 86
4.5.1 The blocking technique - 86
4.5.2 Approximation of S(Rk) by W(Rk) - 87
4.5.3 The remaining terms - 91
4.5.4 End of the proof of Theorem 4.2 - 92
4.6 Annex - 92
5 A nonparametric resampling for non causal random elds and its application to
the texture synthesis 99
5.1 Introduction - 99
5.2 The Markov Mesh Models algorithm - 101
5.2.1 Principle - 101
5.2.2 Consistency results for causal models - 103
5.2.3 An extension to the noncausal case and a convergence rate of Theorem 5.1 . . . 105
5.3 The approach of Paget and Longsta and simulation examples - 108
5.3.1 Paget and Longsta method - 108
5.3.2 Texture synthesis examples - 110
5.4 Annex - 115
5.5 Some tools for the consistency : Continuity results - 118
6 An integer-valued bilinear type model 129
6.1 Introduction - 129
6.2 The model - 133
6.3 Construction of integer-valued models - 134
6.3.1 Basic properties of signed thinning operators - 134
6.3.2 Bilinear model - 134
6.3.3 INLARCH(1) time series model - 135
6.4 Quasi-maximum likelihood estimator in bilinear model - 135
6.4.1 Estimators denition - 138
6.5 QMLE for GINAR(p) processes - 139
6.6 Extended proofs of the results - 140
6.6.1 Proof of Lemma 6.1 - 140
6.6.2 Proof of Theorem 6.2 - 141
6.6.3 Proof of Lemma 6.2 - 142
6.6.4 Proof of Theorem 6.3 - 143
6.6.5 Proof of Theorem 6.4 - 143
6.6.6 Proof of Theorem 6.1 - 146
7 A new smoothed QMLE for AR processes with LARCH errors 155
7.1 Introduction - 155
7.2 Some general results about LARCH models - 157
7.3 Model specication and smoothed QMLE - 158
7.4 Asymptotics of smoothed QMLE for AR-LARCH models - 161
7.5 Choice of the smoothing parameter h - 162
7.6 Numerical illustration - 166
7.7 Proofs - 173
7.7.1 Proof of Lemma 7.1 - 173
7.7.2 Proof of Lemma 7.2 - 173
7.7.3 Proof of Theorem 7.1 - 174
7.7.4 Proof of Theorem 7.2 - 177
7.7.5 Proof of Lemma 7.3 - 180
7.7.6 Proof of Lemma 7.4 - 182
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