Adaptive estimation of mean and volatility functions in (auto-)regressive models

F. Comte and Y. Rozenholc

Stochastic Processes and their Applications, 2002, vol. 97, issue 1, 111-145

Abstract: In this paper, we study the problem of nonparametric estimation of the mean and variance functions b and [sigma]2 in a model: Xi+1=b(Xi)+[sigma](Xi)[var epsilon]i+1. For this purpose, we consider a collection of finite dimensional linear spaces. We estimate b using a mean squares estimator built on a data driven selected linear space among the collection. Then an analogous procedure estimates [sigma]2, using a possibly different collection of models. Both data driven choices are performed via the minimization of penalized mean squares contrasts. The penalty functions are random in order not to depend on unknown variance-type quantities. In all cases, we state nonasymptotic risk bounds in empirical norm for our estimators and we show that they are both adaptive in the minimax sense over a large class of Besov balls. Lastly, we give the results of intensive simulation experiments which show the good performances of our estimator.

Keywords: Nonparametric; regression; Least-squares; estimator; Adaptive; estimation; Autoregression; Variance; estimation; Mixing; processes (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (9)

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