Linear Bayesian Estimation Of Parameters In Pareto Distribution

The Pareto distribution was discovered when studying the problem of income distribution.It is generally used to describe the unbalance of social distribution,such as stock floats,urban population size,and so on.In the non-life insurance actuarial,the two parameters of the Pareto distribution are used to describe the risk characteristic information and deductible information of the policy respectively.Therefore it is appropriate to choose the Pareto distribution to characterize the distribution of claim amount with deductible policy.Due to the non-homogeneous nature of risk,the two parameters of the Pareto distirbution are often regarded as two random variables with a prior distribution in non-life insurance.Therefore,the statistical inference of the parameters of the Pareto distribution falls into the Bayesian framework.In the Bayesian model,how to better integrate the prior distribution into the parameters to improve the accuracy of the parameter estimation is the key content of Bayesian statistical inference.In this paper,we try to establish a Bayesian model of two parameters in the Pareto distribu-tion,and study the moment estimation,maximum likelihood estimation and Bayesian estimation of two parameters,and the statistical properties of these estimates are also be studied.Further-more,according to the credibility theory in non-life insurance actuarial theory,the estimation of shape parameter ? is limited to the linear function of the sample,and the linear Bayesian esti-mation of shape parameter ? is obtained by using the minimum mean square error criterion.In addition,the linear Bayesian estimation of ? has a simply expression,which can be regarded as a weighted average of the maximum likelihood estimation and the aggregate mean,and its mean square error is smaller than other estimates,which is the smallest estimate of the mean square error in the estimation of ? proposed in this paper.Moreover,compared with the Bayesian esti-mation,the linear Bayesian estimation does not depend on the specific prior distribution form of the risk parameter,but only depends on certain moments of the prior distribution,the adjusted maximum likelihood estimation for scale parameters ? has the smallest mean square error.In order to estimate the structural parameters of the linear Bayesian estimation,we establish a Bayesian model of multi-policy contract model for the Pareto distribution.In this model,the maximum likelihood estimation,Bayesian estimation and linear Bayesian estimation of two parameters are obtained.Further,an estimate of the structural parameters is obtained using the empirical linear Bayesian method.The empirical linear Bayesian estimation of the risk parameters is obtained after the estimation of the structural parameters is substituted.Empirical linear Bayesian estimation no longer relies on any information of the prior distribution and can be used in practice directly.Finally,we discuss the multivariate Pareto distribution in the form of random vectors,and give a statistical inference method for the multivariate Pareto distribution.