Abstract :
In this paper we analyze the asymptotic properties of the popular distribution tail
index estimator by Hill (1975) for dependent, heterogeneous processes. We develop
new extremal dependence measures that characterize a massive array of linear, nonlinear,
and conditional volatility processes with long or short memory.We prove that
the Hill estimator is weakly and uniformly weakly consistent for processes with extremes
that form mixingale sequences and asymptotically normal for processes with
extremes that are near epoch dependent (NED) on some arbitrary mixing functional.
The extremal persistence assumptions in this paper are known to hold for mixing,
Lp-NED, and some non-Lp-NED processes, including ARFIMA, FIGARCH, explosive
GARCH, nonlinear ARMA-GARCH, and bilinear processes, and nonlinear
distributed lags like random coefficient and regime-switching autoregressions.
Finally, we deliver a simple nonparametric estimator of the asymptotic variance
of the Hill estimator and prove consistency for processes with NED extremes.
