On estimation of the exponent of regular variation using a sample with missing observations

Let (Xn) be a sequence of possibly dependent random variables with the same marginal distribution function F, such that 1-F(x)=x-αL(x), α>0, where L(x) is a slowly varying function. In this paper the Hill estimator of the exponent of regular variation based on a sample with missing observations from the sequence (Xn) is considered. The asymptotic consistency was proved under some general conditions. This extends results of Hsing [1991. On tail index estimation using dependent data. Ann. Statist. 19, 1547-1569].