Abstract
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.
Citation
Thomas Mikosch. Tamar Gadrich. Claudia Kluppelberg. Robert J. Adler. "Parameter Estimation for ARMA Models with Infinite Variance Innovations." Ann. Statist. 23 (1) 305 - 326, February, 1995. https://doi.org/10.1214/aos/1176324469
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