Adaptive estimation in autoregression or -mixing regression via model selection
Open Access
- 1 June 2001
- journal article
- Published by Institute of Mathematical Statistics in The Annals of Statistics
- Vol. 29 (3), 839-875
- https://doi.org/10.1214/aos/1009210692
Abstract
We study the problem of estimatingsome unknown regression function in a $\beta$-mixing dependent framework. To this end, we consider some collection of models which are finite dimensional spaces. A penalized least-squares estimator (PLSE) is built on a data driven selected model among this collection. We state non asymptotic risk bounds for this PLSE and give several examples where the procedure can be applied (autoregression, regression with arithmetically $\beta$-mixing design points, regression with mixing errors, estimation in additive frameworks, estimation of the order of the autoregression). In addition we show that under a weak moment condition on the errors, our estimator is adaptive in the minimax sense simultaneously over some family of Besov balls.Keywords
This publication has 29 references indexed in Scilit:
- On nonparametric estimation in nonlinear AR(1)-modelsStatistics & Probability Letters, 1999
- Regression-type inference in nonparametric autoregressionThe Annals of Statistics, 1998
- Inequalities for absolutely regular sequences: application to density estimationProbability Theory and Related Fields, 1997
- Minimum complexity regression estimation with weakly dependent observationsIEEE Transactions on Information Theory, 1996
- Universal approximation bounds for superpositions of a sigmoidal functionIEEE Transactions on Information Theory, 1993
- Asymptotic Optimality for $C_p, C_L$, Cross-Validation and Generalized Cross-Validation: Discrete Index SetThe Annals of Statistics, 1987
- Some Limit Theorems for Empirical ProcessesThe Annals of Probability, 1984
- Universal coding, information, prediction, and estimationIEEE Transactions on Information Theory, 1984
- Selection of the order of an autoregressive model by Akaike's information criterionBiometrika, 1976
- A new look at the statistical model identificationIEEE Transactions on Automatic Control, 1974