Some results on estimation of the tail index of a distribution

The author obtains the rate of strong convergence, mean squared error and optimal choice of the “smoothing parameter” (the sample fraction ) of a tail index estimator which was proposed by the author from Pickands’ estimator, and called modified Pickands’ estimator. The similar results about Hill’s estimator are also obtained, which generalize the corresponding results in [8,9]. Besides, some comparisons between Hill’s estimator and the modified Pickands’ estimator are given.