A simple second-order reduced bias' tail index estimator

In this article, we are interested in the direct estimation of the dominant component of the bias of a classical tail index estimator, such as the Hill estimator, used here for illustration of the procedure. Such an estimated bias is then directly removed from the original estimator. The second-order parameters in the bias are based on a number of top order statistics, larger than the one we should use for the estimation of the tail index gamma, so that there is no change in the asymptotic variance of the new reduced bias' tail index estimator, which is kept equal to the asymptotic variance of the classical original one, contrarily to what happens with most of the reduced bias' estimators available in the literature. The asymptotic distributional behaviour of the proposed estimators of gamma is derived, under a second-order framework, and their finite sample properties are also obtained through Monte Carlo simulation techniques.