The Index Of Heavy-tailed Estimator Based On DPR

In many fields such as insurance claims, meteorology, hydrology, environmental science, telecommunications does not meet the normal distribution assumption. They often show that the characteristics of peak and thick tail. How to depict the tail features, that is the problem of estimation of the tail index of a heavy tailed distribution has been paid much attention. Various estimates for the tail index have been proposed in the literature, the classics are Hill estimator and Pickands estimator, these estimators are based on all the upper order statistics. In fact, sometimes only several largest observations are available for analysis, none of aforementioned methods is applicable.In this paper, firstly, we detailed the definition of heavy-tailed distribution and extreme value theory and regular variable conditions. Secondly, we introduced a few of classical tail index estimation based on DPR’s. We propose a new estimator for the tail index under a setup similar to DPR’s, by using the same information as in DPR’s approach. Observations were divided into several blocks and the estimator of the tail index was constructed from the ratios of the a few of upper order statistics. our new estimators are more efficient than DPR’s. The estimator was proved to be asymptotically normal under the second order condition in extreme value theory, and their Edgeworth expansions were obtained. Whenr=1, the asymptotic variance of our estimator is smaller than that of DPR’s estimator for all r=0and the bias for our new estimator is smaller than hat of DPR’s estimator as well. Hill estimator and our estimator have the same asymptotic variance. When r>1, increasing the value of the number of the observations within each block decreases the asymptotic variance of the estimator, meanwhile costs an increase of the asymptotic bias of the estimator. We also do simulation for these estimators in Frechet model and Pareto model, our estimator was applicable in the case of incomplete date, one can conclude that our estimator and the Hill estimator are comparable and both are better than the DPR’s estimator.