×

Parameter estimation for ARMA models with infinite variance innovations. (English) Zbl 0822.62076

Summary: We consider a standard ARMA process of the form \(\varphi (B)X_ t = \theta (B)Z_ t\), 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 \(\varphi\) 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 [P. Whittle, Ark. Mat. 2, 423-434 (1953; Zbl 0053.410)] 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-\({\mathcal L}^ 2\) 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 \({\mathcal L}^ 2\) case.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12 Asymptotic properties of parametric estimators
62M15 Inference from stochastic processes and spectral analysis
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 0053.410
PDFBibTeX XMLCite
Full Text: DOI