Asymptotic Behaviors Of Estimators Of Risk Measures In Pareto Risk Model

In risk management,risk measures can make full use of existing information to analyze and predict risks.Studying the asymptotic behaviors of estimators of risk measures is helpful to make statistical inference on risk measures,so that venture investors can take corresponding measures to avoid risks in time.In this thesis,we consider the asymptotic behaviors of the estimators of different risk measures in two types of Pareto distribution models,including the strong consistency,asymptotic normality,large deviation principle and moderate deviation principle.The main contents are arranged as follows:The first chapter mainly elaborates the research background of Pareto risk model,Pareto-Gamma risk model and related risk measurements,and gives out our main results.Finally,we list some innovations of this thesis.The second chapter mainly introduces the preparation knowledge of the thesis,including the definitions of value at risk,tail value at risk,expected shortfall and conditional value at risk,as well as some necessary concepts and theorems,including the consistency,central limit theorem,large deviation principle and moderate deviation principle.The third chapter is the main research results of this thesis.This chapter consider the asymptotic behaviors of estimators of risk measures in two types of Pareto distribution models.Firstly,the chapter give the maximum likelihood estimators of value at risk,tail value at risk,expected shortfall and conditional value at risk in Pareto risk model,and obtain the strong consistency,central limit theorem and moderate deviation principle of these estimators.Secondly,the chapter consider the Bayesian estimators of the value at risk and the tail value at risk in Pareto-Gamma risk model,and give out the asymptotic behaviors of the Bayesian estimators,including the strong consistency,asymptotic normality,large deviation principle and moderate deviation principle.The fourth chapter is the stochastic simulations and statistical application of the theoretical conclusions in the third chapter.Firstly,this chapter give the stochastic simulations of the consistency of estimators for four risk measures in Pareto risk model,and observe that the mean square error of risk measures decreases with the increase of sample size,thus verifying the strong consistency of estimators.Secondly,this chapter gives the stochastic simulations of confidence interval and interval coverage of risk measures in Pareto-Gamma risk model.Then,for the estimators of two risk measures in Pareto-Gamma risk model,this chapter gives their standardized histogram and kernel density estimation curve,which basically coincides with the standard normal distribution density function curve,thus verifying the asymptotic normality of the estimators.At the same time,this chapter gives stochastic simulations of the tail probability of the estimators.The simulations show that the tail probability tends to zero at a certain speed when the sample size is large enough,thus verifying the moderate deviation principle of the estimators.Finally,this chapter gives the specific application of the moderate deviation principle,which is for estimator of the tail value at risk in Pareto-Gamma risk model,in statistical hypothesis testing.In the end,the main work of this thesis is briefly summarized,and some problems worthy of further study are given.