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

Hakim Ouadjed, Mami Tawfiq Fawzi

We consider estimating the extremal index of a maximum autoregressive process of order one under the assumption that the distribution of the innovations has a regularly varying tail at infinity. We establish the asymptotic normality of the new estimator using the extreme quantile approach, and its performance is illustrated in a simulation study. Moreover, we compare, in terms of bias and mean squared error, our estimator with the estimator 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-normal estimators for the extremal index, {Econometrics}, 3 (2015), 633--653].