On Tail Index Estimation Using Dependent,Heterogenous Data

In this paper we analyze the asymptotic properties of the popularly used distribution tail estimator by B. Hill (1975), for heavy-tailed heterogenous, dependent processes. We prove the Hill estimator is weakly consistent for functionals of mixingales and L1-approximable processes with regularly varying tails, covering ARMA, GARCH, and many IGARCH and FIGARCH processes. Moreover, for functionals of processes near epoch-dependent on a mixing process, we prove a Gaussian distribution limit exists. In this case, as opposed to all existing prior results in the literature, we do not require the limiting variance of the Hill estimator to be bounded, and we develop a Newey-West kernel estimator of the variance. We expedite the theory by defining "extremal mixingale" and "extremal NED" properties to hold exclusively in the extreme distribution tails, disbanding with dependence restrictions in the non-extremal support, and prove a broad class of linear processes are extremal NED. We demonstrate that for greater degrees of serial dependence more tail information is required in order to ensure asymptotic normality, both in theory and practice.