Weak Dependence: With Examples and Applications - With Examples and Applications

Time series and random ?elds are main topics in modern statistical techniques. They are essential for applications where randomness plays an important role. Indeed, physical constraints mean that serious modelling cannot be done - ing only independent sequences. This is a real problem because asymptotic properties are not always known in this case. Thepresentworkisdevotedtoprovidingaframeworkforthecommonlyused time series. In order to validate the main statistics, one needs rigorous limit theorems. In the ?eld of probability theory, asymptotic behavior of sums may or may not be analogous to those of independent sequences. We are involved with this ?rst case in this book. Very sharp results have been proved for mixing processes, a notion int- duced by Murray Rosenblatt [166]. Extensive discussions of this topic may be found in his Dependence in Probability and Statistics (a monograph published by Birkhau ¨ser in 1986) and in Doukhan (1994) [61], and the sharpest results may be found in Rio (2000)[161]. However, a counterexample of a really simple non-mixing process was exhibited by Andrews (1984) [2]. The notion of weak dependence discussed here takes real account of the available models, which are discussed extensively. Our idea is that robustness of the limit theorems with respect to the model should be taken into account. In real applications, nobody may assert, for example, the existence of a density for the inputs in a certain model, while such assumptions are always needed when dealing with mixing concepts.

Sommario

Weak dependence.- Models.- Tools for non causal cases.- Tools for causal cases.- Applications of strong laws of large numbers.- Central Limit theorem.- Donsker Principles.- Law of the iterated logarithm (LIL).- The Empirical process.- Functional estimation.- Spectral estimation.- Econometric applications and resampling.

Riassunto

Time series and random ?elds are main topics in modern statistical techniques. They are essential for applications where randomness plays an important role. Indeed, physical constraints mean that serious modelling cannot be done - ing only independent sequences. This is a real problem because asymptotic properties are not always known in this case. Thepresentworkisdevotedtoprovidingaframeworkforthecommonlyused time series. In order to validate the main statistics, one needs rigorous limit theorems. In the ?eld of probability theory, asymptotic behavior of sums may or may not be analogous to those of independent sequences. We are involved with this ?rst case in this book. Very sharp results have been proved for mixing processes, a notion int- duced by Murray Rosenblatt [166]. Extensive discussions of this topic may be found in his Dependence in Probability and Statistics (a monograph published by Birkhau ¨ser in 1986) and in Doukhan (1994) [61], and the sharpest results may be found in Rio (2000)[161]. However, a counterexample of a really simple non-mixing process was exhibited by Andrews (1984) [2]. The notion of weak dependence discussed here takes real account of the available models, which are discussed extensively. Our idea is that robustness of the limit theorems with respect to the model should be taken into account. In real applications, nobody may assert, for example, the existence of a density for the inputs in a certain model, while such assumptions are always needed when dealing with mixing concepts.

Testo aggiuntivo

From the reviews:

"I appreciate this book as a very welcome and thorough discussion of the actual state-of-the art in the modeling of dependence structures. It provides a large number of motivating examples and applications, rigorous proofs, and valuable intuitions for the willing and mathematically well-trained reader with essential prior knowledge of the mathematical prerequisites of weak dependence … . It is … the book to those researchers already aware of the necessity of the methods discussed here." (Harry Haupt, Advances in Statistical Analysis, Vol. 93, 2009)

"This book … provides a detailed description of the notion of weak dependence as well as properties and applications. … Overall the book is neatly written … . the book is very rich in its material as it contains earlier works on dependence and … show a lot of applications of the theory. It also contains a large number of examples and expositions of the idea of weak dependence in models … which provide good insight." (Dimitris Karlis, Zentralblatt MATH, Vol. 1165, 2009)

Relazione

From the reviews:

"I appreciate this book as a very welcome and thorough discussion of the actual state-of-the art in the modeling of dependence structures. It provides a large number of motivating examples and applications, rigorous proofs, and valuable intuitions for the willing and mathematically well-trained reader with essential prior knowledge of the mathematical prerequisites of weak dependence ... . It is ... the book to those researchers already aware of the necessity of the methods discussed here." (Harry Haupt, Advances in Statistical Analysis, Vol. 93, 2009)
"This book ... provides a detailed description of the notion of weak dependence as well as properties and applications. ... Overall the book is neatly written ... . the book is very rich in its material as it contains earlier works on dependence and ... show a lot of applications of the theory. It also contains a large number of examples and expositions of the idea of weak dependence in models ... which provide good insight." (Dimitris Karlis, Zentralblatt MATH, Vol. 1165, 2009)

Dettagli sul prodotto

• Autori: Jérom Dedecker, Jérome Dedecker, Jérôme Dedecker, Pau Doukhan, Paul Doukhan, Gabriel Lang, Gabriel et al Lang, José Rafael Leon, Sana Louhichi, Clémentine Prieur
• Editore: Springer, Berlin
• Lingue: Inglese
• Formato: Tascabile
• Pubblicazione: 20.08.2007
• EAN: 9780387699516
• ISBN: 978-0-387-69951-6
• Pagine: 322
• Dimensioni: 160 mm x 232 mm x 19 mm
• Peso: 536 g
• Illustrazioni: XIV, 322 p.
• Serie: Lecture Notes in Statistics; Lecture Notes in Statistics
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Analisi; C, Statistics, Mathematics and Statistics, Statistical Theory and Methods, Resampling;fitting;parametric statistics;statistics