Minimum-Variance Reduced-Bias Tail Index and High Quantile Estimation

Abstract

Heavy tailed-models are quite useful in many fields, like insurance, finance, telecommunications, internet traffic, among others, and it is often necessary to estimate a high quantile, i.e., a value that is exceeded with a probability p, small. The semiparametric estimation of this parameter relies essentially on the estimation of the tail index, the primary parameter in statistics of extremes. Classical semi-parametric estimators of extreme parameters show usually a severe bias and are known to be very sensitive to the number k of top order statistics used in the estimation. For k small they have a high variance, and for large k a high bias. Recently, new second-order “shape” and “scale” estimators allowed the development of second-order reduced-bias estimators, which are much less sensitive to the choice of k. Here we shall study, under a third order framework, minimum-variance reduced-bias (MVRB) tail index estimators, recently introduced in the literature, and dependent on an adequate estimation of second order parameters. The improvement comes from the asymptotic variance, which is kept equal to the asymptotic variance of the classical Hill estimator, provided that we estimate the second order parameters at a level of a larger order than the level used for the estimation of the first order parameter. The use of those MVRB tail index estimators enables us to introduce new classes of reduced-bias high quantile estimators. These new classes are compared among themselves and with previous ones through the use of a small-scale Monte Carlo simulation.