The Annals of Statistics

Whittle estimator for finite-variance non-Gaussian time series with long memory

Liudas Giraitis and Murad S. Taqqu

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We consider time series $Y_t = G(X_t)$ where $X_t$ is Gaussian with long memory and $G$ is a polynomial. The series $Y_t$ may or may not have long memory. The spectral density $g_\theta(x)$ of $Y_t$ is parameterized by a vector $\theta$ and we want to estimate its true value $\theta_0$ . We use a least-squares Whittle-type estimator $\hat{\theta}_N$ for $\theta_0$, based on observations $Y_1,\dots,Y_N$. If $Y_t$ is Gaussian, then $\sqrt{N}(\hat{\theta}_N-\theta_0)$ converges to a Gaussian distribution. We show that for non-Gaussian time series $Y_t$ , this $\sqrt{N}$ consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if $Y_t$ has short memory.

Article information

Ann. Statist., Volume 27, Number 1 (1999), 178-203.

First available in Project Euclid: 5 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62F10: Point estimation
Secondary: 60G18: Self-similar processes

Hermite polynomials non-central limit theorem long-range dependence quadratic forms time series


Giraitis, Liudas; Taqqu, Murad S. Whittle estimator for finite-variance non-Gaussian time series with long memory. Ann. Statist. 27 (1999), no. 1, 178--203. doi:10.1214/aos/1018031107.

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