Heavy-tailed Phenomena, Heavy-tailed Distribution And Heavy-tailed Index Estimators

Many studies show that financial asset prices, insurance claims, network traffic and many phenomena of human behavior does not meet the normal distribution assumption, but are subject to heavy-tailed distribution. There is a very important significance for study of risks laws, risk prediction and promoting financial security to run efficiently and steadily to study the heavy tailed distribution. How to estimate the heavy tail index accurately has been the focus of scholars' attention.First introduced the heavy-tailed Phenomena and definition of heavy-tailed distribu-tion, second introduced several kinds of classical methods to estimate the heavy-tailed in-dex and some representative methods to select the heavy-tailed threshold, third introduced the full asymptotic distributional properties of Hill estimator and several reduced-bias esti-mators proposed in recent years. At last we provided an analytical method, named Simple and Optimized Estimator(SOE), and another method for large samples, both to choose threshold for heavy-tailed index estimation. Then we proceed to an intensive computer simulation that enables us to find, through Monte-Carlo techniques, the SOE method is accurate and robust as well as Sum-plot method and Bootstrap method. The SOE method seems to have a slight advantage in terms of accuracy, the calculation method is simple, but not be affected by the types of the heavy-tailed distributions and the scope of the index, the Hill estimator based on the SOE method is still consistent whenever the distributions are Pareto-type models. Besides, we propose two classes of new heavy tailed index estima-tors under a third order framework, as well as the full asymptotic distributional properties of the two classes of estimators are derived. The biases of the estimators proposed by us are smaller in the premise that the variance equals toγ2, not only asymptotically, but also for finite samples through Monte Carlo techniques.