Hill's estimator for the tail index of an ARMA model

Abstract

This paper investigates Hill's estimator for the tail index of an ARMA model with i.i.d. residuals. Based on the estimated residuals, it is shown that Hill's estimator is asymptotically normal. This method can achieve a smaller asymptotic variance than applying Hill's estimator to the original data. These results are the same as those in Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) for an AR model. However, Resnick and Starica (Commun. Statist.—Stochastic Models 13 (4) (1997) 703) imposed one more condition on the choice of sample fraction than the i.i.d. case. This condition is removed in this paper so that data-driven methods for choosing optimal sample fraction based on i.i.d. data can be applied to our case. As an auxiliary theorem, we establish the weak convergence of the tail empirical process of the estimated residuals, which may be of independent interest.