Tail Parameters Estimation And Theoretical Derivation Of Heavy-tailed Distribution Based On The Extreme Value Theory

In light of the increasing number of events which result in catastrophic loss and occur with extremely small probability,we are increasingly concerned about the way to deal with these events.If we are able to accurately predict these extreme events,we can make the appropriate preparations before they happen.This requires accurate estimation of model parameters.At present,the generalized Pareto distribution is commonly used to model the extreme value data.The value of the extreme value index of the distribution can be used to measure the possibility which the extreme events happen.Before the extreme value index is estimated an appropriate threshold has to be selected.By combining the selected threshold with the parameter estimation method,the extreme value index of the generalized Pareto distribution can be estimated.This thesis discusses the method of threshold selection and the method of extreme value index estimation for the generalized Pareto distribution.An unconditional threshold selection method is proposed for the generalized Pareto distribution.This thesis use the mean excess plot as a preliminary analysis for possible candidates of the threshold.Note that the mean excess plot can not choose a unique threshold.Using the distribution function of the GP distribution,the likelihood ratio statistic and score statistic are calculated,and the two statistics are respectively analysed with the mean excess plot to select the appropriate threshold.The mean excess plot can be used to find out a possible range of the threshold candidates.Based on the maximum value of the likelihood ratio statistic and the score statistic in this range,the appropriate threshold value can be selected.The method of selecting the threshold not only overcomes the shortcomings of the mean excess plot,but also increases the accuracy of threshold selection.In parameter estimation,the MLE method is used to estimate the parameters of the generalized Pareto distribution.The maximum likelihood method is based on the principle of maximum likelihood which maximize the log-likelihood function defined by logarithm of the joint probability density function of the generalized Pareto distribution.When the maximum value of the log-likelihood function is reached,the value of the extreme value index is the estimated extreme value index.By maximizing the log-likelihood function at each of the possible threshold candidates,the unconditional tail inference picks the one which gives the maximum value of the log-likelihood function.Hence,the resulting parameter estimates maximize the information available from the data.The unconditional tail inference is supplemented by an real data example.The medical claims data of SOA in 1991 is a typical example of the heavy-tailed distribution which can be modeled by the generalized Pareto distribution.The unconditional threshold selection method can select the optimal threshold of SOA medical claims data and avoid threshold uncertainty.Based on the selected threshold,the extreme value index can be estimated by the maximum likelihood method.The extreme value index can be accurately estimated in an unconditional way.