The Generalized Pareto Distribution

Pareto distribution appeared 150 years ago. With the development of society, Pareto distribution has been improved and extended to several kinds of Pareto distribution, which formed Pareto Distribution Family. Pareto Distribution Family deserves so many researchers' attention because of its' excellent properties. This paper introduces the development of Pareto Distribution Family and it's application in Economics, Sociology, Euthenics and Actuarial Statistics. Two of Pareto Distribution Family have often been used by actuary. So, it is significative to research Pareto Distribution Family.Probability density function, moment, mode, Fish information matrix, order statistics and parameter estimator is introduced in the second part .The relation among the Pareto Distribution Family inside is introduced too.Such we can understand the development of Pareto Distribution Family. And we can know the relation between the distributions of Pareto Distribution Family and uniform distribution, F distribution, Beta distribution, Z distribution.Generalized Pareto distribution is introduced in the third part which is the keystone of this paper. The difference between generalized Pareto distribution and the Pareto distribution mentioned in the paper of Lihaifen is the bound of variable is extended, parameter is added to three. Based on the probability density function and other characters, the moment and distribution function of generalized Pareto distribution are calculated. The parameter estimators including moment estimator, maximum likelihood estimator and interval estimator are analyzed. Based on the normal infer, some characters of maximum likelihood estimator are researched. Order statistics and its' distributions of generalized Pareto distribution are introduced. Likelihood-ratio test is a important parameter test method which it is applied to generalized Pareto distribution. Generalized Pareto distribution play key role in practical cases, such as income data exceed a fixed bound , hurricane loss and so on. It is important to research generalized Pareto distribution.