Moderate deviations for the Durbin–Watson statistic related to the first-order autoregressive process

¹ Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, Avenue des Landais, 63177 Aubière, France
Valere.Bitsekipenda@math.univ-bpclermont.fr
² Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, Avenue des Landais, 63177 Aubière, France
Hacene.Djellout@math.univ-bpclermont.fr
³ Université Bordeaux 1, Institut de Mathématiques de Bordeaux, UMR 5251, and INRIA Bordeaux, team ALEA, 200 Avenue de la Vieille Tour, 33405 Talence cedex, France
Frederic.Proia@inria.fr

Received: 31 August 2012
Revised: 26 February 2013

Abstract

The purpose of this paper is to investigate moderate deviations for the Durbin–Watson statistic associated with the stable first-order autoregressive process where the driven noise is also given by a first-order autoregressive process. We first establish a moderate deviation principle for both the least squares estimator of the unknown parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. It enables us to provide a moderate deviation principle for the Durbin–Watson statistic in the case where the driven noise is normally distributed and in the more general case where the driven noise satisfies a less restrictive Chen–Ledoux type condition.