ON TAIL INDEX ESTIMATION USING DEPENDENT DATA

Let X1, X2, ... be possibly dependent random variables having the same marginal distribution. Consider the situation where FBAR(x): = P[X1 > x] is regularly varying at infinity with an unknown index -alpha < 0 which is to be estimated. In the i.i.d. setting, it is well known that Hill's estimator is consistent for alpha-1, and is asymptotically normally distributed. It is the purpose of this paper to demonstrate that such properties of Hill's estimator extend considerably beyond the independent setting. In addition to some basic results derived under very general conditions, the case where the observations are strictly stationary and satisfy a certain mixing condition is considered in detail. Also a finite moving average sequence is studied to illustrate the results.