Estimating Tail Decay for Stationary Sequences via Extreme Values

A. Zeevi and P. W. Glynn

Advances in Applied Probability, 198-226 (2004)

We study estimation of the tail decay parameter of the marginal distribution corresponding to a discrete time, real valued stationary stochastic process. Assuming that the underlying process is short-range dependent, we investigate properties of estimators of the tail decay parameter which are based on the maximal extreme value of the process observed over a sampled time interval. These estimators only assume that the tail of the marginal distribution is roughly exponential, plus some modest "mixing" conditions. Consistency properties of these estimators are established, as well as rates of convergence and robustness in a minimax sense. We also provide some discussion on estimating the pre-exponent, when a more refined tail asymptotic is assumed. Properties of a certain moving-average variant of the extremal-based estimator are investigated as well. In passing, we also characterize the precise dependence (mixing) assumptions that support almost sure limit theory for normalized extreme values and related first passage times in stationary sequences.