Semi parametric estimation of extremal index for ARMAX process with infinite variance

Abstract

We consider estimating the extremal index of a maximumautoregressive process of order one under the assumption that thedistribution of the innovations has a regularly varying tail atinfinity. We establish the asymptotic normality of the newestimator using the extreme quantile approach, and its performanceis illustrated in a simulation study. Moreover, we compare, interms of bias and mean squared error, our estimator with theestimator of Ferro and Segers [Inference for clusters of extreme values, J. Royal Stat. Soc.,Ser. B, {65} (2003), 545–556] and Olmo [A new family of consistent and asymptotically-normalestimators for the extremal index, {Econometrics}, 3 (2015), 633–653].