Tail index estimation in the presence of long-memory dynamics

Most tail index estimators are formulated under assumptions of weak serial dependence, but nevertheless are applied in practice to long-range dependent time series data. This issue arises because for many time series found in teletraffic and financial econometric applications, both heavy tails and long memory are prevalent features. For a certain class of Heavy-Tail Long-Memory (HTLM) processes, McElroy and Politis (2007a) and Jach et al. (2011) found that the probabilistic behavior of the sample mean depends delicately on the interplay of the tail index and the long memory parameter. In contrast, results in Kulik and Soulier (2011) indicate that the sample quantiles for a related HTLM process are unaffected by long-range dependence. Motivated by these results, we undertake an extensive numerical study to compare the finite-sample performance of several tail index estimators–both those based on sample quantiles, such as the Hill and DEdH (Hill (1975) and Dekkers et al. (1989)) as well as those based on moments, e.g. Meerschaert and Scheffler (1998)–in the HTLM context. Our results largely confirm and expand those of Kulik and Soulier (2011), in that the Hill and DEdH estimators perform well despite the presence of long memory.