A Wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series

Éric Moulines (1) , François Roueff (1) , Murad S. Taqqu (2)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 BU - Boston University [Boston]

We consider a time series $X=\{X_k,\,k\in\mathbb{Z}\}$ with memory parameter $d\in\mathbb{R}$. This time series is either stationary or can be made stationary after differencing a finite number of times. We study the ``Local Whittle Wavelet Estimator'' of the memory parameter $d$. This is a wavelet-based semiparametric pseudo-likelihood maximum method estimator. The estimator may depend on a given finite range of scales or on a range which becomes infinite with the sample size. We show that the estimator is consistent and rate optimal if $X$ is a linear process and is asymptotically normal if $X$ is Gaussian.

Mots clés: long memory; semiparametric estimation; Wavelet analysis

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Éric Moulines, François Roueff, Murad S. Taqqu. A Wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series. Annals of Statistics, 2008, 36 (4), pp.1925-1956. ⟨10.1214/07-AOS527⟩. ⟨hal-00016446v4⟩