The Second Order Parameters’ Estimator Of Heavy-tailed Distribution

A lot of studies show that heavy-tailed distribution appears often in fields like finance, insurance, meteorology, hydrology, environmental, sociology and so on. They often show the characteristics of peak and thick tail, namely, have comparatively thick tail compared with normal distribution. How to depict the tail features, that is how to estimate the tail index of heavy-tailed distribution （Professor Liu Weiqi said heavy-tailed index）, becomes the focus of academic concern. However, the second order parameter which is closely related to heavy-tailed index, doesn’t allow to ignore. So statisticians of extreme value theory attach importance to the second order parameter.In the paper, firstly, we detailed the definition of heavy-tailed distribution and theo-retical basis-extreme value theory and regular variable conditions. Secondly, we review a few of classical methods of second order parameters’ estimator, and based on the statistics Mn（α）（k）, we present a new class of second order parameters’ estimator. The consistency of the second order parameter is studied under the second order condition in extreme value theory, and asymptotic normality is achieved under the third order condition. At last, we discuss the selection of the parameter α considering the asymptotic bias components, the asymptotic standard deviations components and the root of asymptotic mean squared er-rors. The results show that the smaller α is, the better the performance of new estimator is. Meanwhile, the new estimator, the estimator ρGM provided by Gomes et al.（2002） and the estimator ρFA（0） provided by Fraga Alves et al.（2003） for the two aspects of large sample behavior and small sample behavior is done by means of Monte-Carlo simulation techniques in Frechet and Burr models.The main conclusions are as follows.For the large sample behavior, the estimator provided in the paper works very well under consideration the asymptotic bias, the asymptotic standard deviations and the root of asymptotic mean squared errors.For the small sample behavior, sample mean values and mean square errors of second order parameters’estimators are simulated at the optimal level. The smaller a is, the better p（?）^a works in Frechet（1,-1） model. For Burr（1,p） model, the performance of p（?）^α is better with the increase of a when p values in the range-1to0; for p<-1, the estimator p（?）^α shows the better performance as a gets smaller.