Dependence in Probability and Statistics

by Doukhan, Paul; Lang, Gabriel; Surgailis, Donatas; Teyssiere, Gilles

• ISBN13: 9783642141034
• ISBN10: 364214103X
• Format: Paperback
• Copyright: 2010-10-12
• Publisher: Springer Nature

Summary

This volume collects recent works on weakly dependent, long-memory and multifractal processes and introduces new dependence measures for studying complex stochastic systems. Other topics include the statistical theory for bootstrap and permutation statistics for infinite variance processes, the dependence structure of max-stable processes, and the statistical properties of spectral estimators of the long memory parameter. The asymptotic behavior of Fejér graph integrals and their use for proving central limit theorems for tapered estimators are investigated. New multifractal processes are introduced and their multifractal properties analyzed. Wavelet-based methods are used to study multifractal processes with different multiresolution quantities, and to detect changes in the variance of random processes. Linear regression models with long-range dependent errors are studied, as is the issue of detecting changes in their parameters.

Table of Contents

Permutation and bootstrap statistics under infinite variance	p. 1
Introduction	p. 1
Some general sampling theorems	p. 2
Application to change point detection	p. 10
References	p. 19
Max-Stable Processes: Representations, Ergodic Properties and Statistical Applications	p. 21
Introduction	p. 21
Representations of Max-Stable Processes	p. 24
Ergodic Properties of Stationary Max-stable Processes	p. 29
Examples and Statistical Applications	p. 32
Ergodic Properties of Some Max-Stable Processes	p. 32
Estimation of the Extremal Index	p. 35
References	p. 40
Best attainable rates of convergence for the estimation of the memory parameter	p. 43
Introduction	p. 43
Lower bound	p. 45
Upper bound	p. 47
Bandwidth selection	p. 49
Technical results	p. 51
References	p. 56
Harmonic analysis tools for statistical inference in the spectral domain	p. 59
Introduction	p. 59
Motivation	p. 61
Main result	p. 64
Applications and discussion	p. 67
References	p. 70
On the impact of the number of vanishing moments on the dependence structures of compound Poisson motion and fractional Brownian motion in multifractal time	p. 71
Motivation	p. 72
Infinitely divisible processes	p. 74
Infinitely divisible cascade	p. 74
Infinitely divisible motion	p. 76
Fractional Brownian motion in multifractal time	p. 78
Multiresolution quantities and scaling parameter estimation	p. 78
Multiresolution quantities	p. 78
Scaling parameter estimation procedures	p. 80
Dependence structures of the multiresolution coefficients: analytical study	p. 80
Correlation structures for increment and wavelet coefficients	p. 81
Higher order correlations for increments	p. 83
Role of the order of the increments	p. 85
Dependence structures of the multiresolution coefficients: Conjectures and numerical studies	p. 85
Conjectures	p. 85
Numerical simulations	p. 86
Discussions and conclusions on the role of the number of vanishing moments:	p. 89
Proofs	p. 90
A key lemma	p. 90
Proof of Theorem 4.1	p. 91
Proof of Theorem 4.2	p. 93
Proof of Proposition 0.6	p. 94
Proof of Proposition 0.7	p. 95
Proof of Proposition 0.8	p. 99
References	p. 99
Multifractal scenarios for products of geometric Ornstein-Uhlenbeck type processes	p. 103
Introduction	p. 103
Multifractal products of stochastic processes	p. 104
Geometric Ornstein-Uhlenbeck processes	p. 108
Multifractal Ornstein-Uhlenbeck processes	p. 113
Log-tempered stable scenario	p. 113
Log-normal tempered stable scenario	p. 116
References	p. 120
A new look at measuring dependence	p. 123
Introduction	p. 123
Bivariate dependence	p. 125
Global dependence measures	p. 126
Local dependence measures	p. 130
Connections with reliability theory	p. 133
Multivariate dependence	p. 135
Moment inequalities and limit theorems	p. 137
References	p. 139
Robust regression with infinite moving average errors	p. 143
Introduction	p. 143
S-estimators	p. 144
S-estimators' Asymptotic Behavior	p. 145
Weak Convergence of estimators	p. 146
Proof of Proposition 1	p. 150
Discussion	p. 156
References	p. 156
A note on the monitoring of changes in linear models with dependent errors	p. 159
Introduction	p. 159
The testing procedure	p. 160
Examples	p. 163
Linear models with NED regressors	p. 163
Linear models with asymptotically M-dependent errors	p. 164
Monitoring strongly mixing AR models	p. 165
Proofs	p. 167
References	p. 173
Testing for homogeneity of variance in the wavelet domain	p. 175
Introduction	p. 176
The wavelet transform of K-th order difference stationary processes	p. 178
Asymptotic distribution of the W2-CUSUM statistics	p. 180
The single-scale case	p. 180
The multiple-scale case	p. 187
Test statistics	p. 190
Power of the W2-CUSUM statistics	p. 192
Power of the test in single scale case	p. 192
Power of the test in multiple scales case	p. 196
Some examples	p. 198
References	p. 204
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