The Annals of Statistics

Parameter Estimation for ARMA Models with Infinite Variance Innovations

Thomas Mikosch, Tamar Gadrich, Claudia Kluppelberg, and Robert J. Adler

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We consider a standard ARMA process of the form $\phi(B)X_t = \theta(B)Z_t$, where the innovations $Z_t$ belong to the domain of attraction of a stable law, so that neither the $Z_t$ nor the $X_t$ have a finite variance. Our aim is to estimate the coefficients of $\phi$ and $\theta$. Since maximum likelihood estimation is not a viable possibility (due to the unknown form of the marginal density of the innovation sequence), we adopt the so-called Whittle estimator, based on the sample periodogram of the $X$ sequence. Despite the fact that the periodogram does not, a priori, seem like a logical object to study in this non-$\mathscr{L}^2$ situation, we show that our estimators are consistent, obtain their asymptotic distributions and show that they converge to the true values faster than in the usual $\mathscr{L}^2$ case.

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Ann. Statist., Volume 23, Number 1 (1995), 305-326.

First available in Project Euclid: 11 April 2007

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Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M15: Spectral analysis 62E20: Asymptotic distribution theory 62F10: Point estimation

Stable innovations ARMA process periodogram Whittle estimator parameter estimation


Mikosch, Thomas; Gadrich, Tamar; Kluppelberg, Claudia; Adler, Robert J. Parameter Estimation for ARMA Models with Infinite Variance Innovations. Ann. Statist. 23 (1995), no. 1, 305--326. doi:10.1214/aos/1176324469.

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