A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator

Abstract

For heavy tails, with a positive tail index γ , classical tail index estimators, like the Hill estimator, are known to be quite sensitive to the number of top order statistics k used in the estimation, whereas second-order reduced-bias estimators show much less sensitivity to changes in k . In the recent minimum-variance reduced-bias (MVRB) tail index estimators, the estimation of the second order parameters in the bias has been performed at a level k 1 of a larger order than that of the level k at which we compute the tail index estimators. Such a procedure enables us to keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator, for all k at which we can guarantee the asymptotic normality of the Hill statistics. These values of k , as well as larger values of k , will also enable us to guarantee the asymptotic normality of the reduced-bias estimators, but, to reach the minimal mean squared error of these MVRB estimators, we need to work with levels k and k 1 of the same order. In this note we derive the way the asymptotic variance varies as a function of q , the finite limiting value of k / k 1 , as the sample size n increases to infinity.