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

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.

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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

Information

Published: February, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0822.62076
MathSciNet: MR1331670
Digital Object Identifier: 10.1214/aos/1176324469

Subjects:
Primary: 62M10
Secondary: 62E20, 62F10, 62M15

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 1 • February, 1995
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