Weighted Empirical Processes in Dynamic Nonlinear Models

The role of the weak convergence technique via weighted empirical processes has proved to be very useful in advancing the development of the asymptotic theory of the so called robust inference procedures corresponding to non-smooth score functions from linear models to nonlinear dynamic models in the 1990's. This monograph is an ex panded version of the monograph Weighted Empiricals and Linear Models, IMS Lecture Notes-Monograph, 21 published in 1992, that includes some aspects of this development. The new inclusions are as follows. Theorems 2. 2. 4 and 2. 2. 5 give an extension of the Theorem 2. 2. 3 (old Theorem 2. 2b. 1) to the unbounded random weights case. These results are found useful in Chapters 7 and 8 when dealing with ho moscedastic and conditionally heteroscedastic autoregressive models, actively researched family of dynamic models in time series analysis in the 1990's. The weak convergence results pertaining to the partial sum process given in Theorems 2. 2. 6 . and 2. 2. 7 are found useful in fitting a parametric autoregressive model as is expounded in Section 7. 7 in some detail. Section 6. 6 discusses the related problem of fit ting a regression model, using a certain partial sum process. Inboth sections a certain transform of the underlying process is shown to provide asymptotically distribution free tests. Other important changes are as follows. Theorem 7. 3.

Sommario

1 Introduction.- 1.1 Weighted Empirical Processes.- 1.2 M-, R- and Scale Estimators.- 1.3 M.D. Estimators & Goodness-of-fit Tests.- 1.4 R.W.E. Processes and Dynamic Models.- 2 Asymptotic Properties of W.E.P.'s.- 2.1 Introduction.- 2.2 Weak Convergence.- 2.3 AUL of Residual W.E.P.'s.- 2.4 Some Additional Results for W.E.P.'S.- 3 Linear Rank and Signed Rank Statistics 69.- 3.1 Introduction.- 3.2 AUL of Lin ear Rank Statistics.- 3.3 AUL of Linear Signed Rank Statistics.- 3.4 Weak Convergence of Rank and Signed Rank W.E.P.'s.- 4 M, R and Some Scale Estimators.- 4.1 Introduction.- 4.2 M-Estimators.- 4.3 Distributions of Some Scale Estimators.- 4.4 R-Estimators.- 4.5 Est imation of Q(f).- 5 Minimum Distance Estimators.- 5.1 Introduction.- 5.2 Definitions of M.D. Estimators.- 5.3 Finite Sample Properties.- 5.4 A General M.D. Estimator.- 5.5 Asymptotic Uniform Quadraticity.- 5.6 Distributions, Efficiency & Robustness.- 6 Goodness-of-fit Tests in Regression.- 6.1 Introducti on.- 6.2 The Supremum Distance Tests.- 6.3 L2-Distance Tests.- 6.4 Testing with Unknown Scale.- 6.5 Testing for the Symmetry of the Errors.- 6.6 Regression Model Fitting.- 7 Autoregression.- 7.1 Introduction.- 7.2 AUL of Wh and Fn.- 7.3 GM- and GR- Estimators.- 7.4 Minimum Distance Estimation.- 7.5 Autoregression Quantiles and Rank Scores.- 7.6 Goodness-of-fit Testing for F.- 7.7 Autoregressive Model Fitting.- 8 Nonlinear Autoregression 358.- 8.1 Introduction.- 8.2 AR Models.- 8.3 ARCH Models.- Lectures Notes in Statistics.

Info autore

Hira L. Koul is a professor of statistics at Michigan State University. He is a Fellow of the IMS and an Elected Member of the International Statistical Institute. He was awarded the presti- gious Humboldt Research Award for Senior Researchers in 1995. He has been on the editorial boards of the Annals of Statistics, Sankhya, and J. Indian Statistical Association. Currently he is a Coordinating Editor of the Journal of Statistical Planning and Inference, and an Associate Editor of Statistics and Probability Letters.

Riassunto

The role of the weak convergence technique via weighted empirical processes has proved to be very useful in advancing the development of the asymptotic theory of the so called robust inference procedures corresponding to non-smooth score functions from linear models to nonlinear dynamic models in the 1990's. This monograph is an ex­ panded version of the monograph Weighted Empiricals and Linear Models, IMS Lecture Notes-Monograph, 21 published in 1992, that includes some aspects of this development. The new inclusions are as follows. Theorems 2. 2. 4 and 2. 2. 5 give an extension of the Theorem 2. 2. 3 (old Theorem 2. 2b. 1) to the unbounded random weights case. These results are found useful in Chapters 7 and 8 when dealing with ho­ moscedastic and conditionally heteroscedastic autoregressive models, actively researched family of dynamic models in time series analysis in the 1990's. The weak convergence results pertaining to the partial sum process given in Theorems 2. 2. 6 . and 2. 2. 7 are found useful in fitting a parametric autoregressive model as is expounded in Section 7. 7 in some detail. Section 6. 6 discusses the related problem of fit­ ting a regression model, using a certain partial sum process. Inboth sections a certain transform of the underlying process is shown to provide asymptotically distribution free tests. Other important changes are as follows. Theorem 7. 3.

Dettagli sul prodotto

• Autori: H. L. Koul, Hira L Koul, Hira L. Koul
• Con la collaborazione di: Hira L. Koul (Editore)
• Editore: Springer, Berlin
• Lingue: Inglese
• Formato: Tascabile
• Pubblicazione: 31.07.2002
• EAN: 9780387954769
• ISBN: 978-0-387-95476-9
• Pagine: 425
• Peso: 636 g
• Illustrazioni: XVII, 425 p. 1 illus.
• Serie: Lecture Notes in Statistics; Lecture Notes in Statistics
• Categoria: Scienze naturali, medicina, informatica, tecnica > Matematica > Teoria delle probabilità, stocastica, statistica matematica