On The Use Of Posterior Regret Γ-minimax Actions To Obtain Bayesian Premiums

Calculating premium is an important work in insurance. Many methods can be usedto calculate insurance premium and the Bayesian premium theory is one of the mostwidely used in present. In this paper, we mainly focus on the use of posterior regretΓ-minimax actions to obtain Bayesian premiums.Computing Bayesian premium requires the use of a prior distribution that theunknown risk parameter. In practice, prior information is often vague and the elicitationof a prior is quite difficult. According to robust Bayesian methodology, the prioruncertainty can be modeled by specifying a classΓ of prior distributions and computethe ranges of Bayesian actions when the prior ranges overΓ. By using robust Bayesianmethodology, the statistician now has a range of possible actions; perhaps it is unclearas to which of these are the optimal procedures. There are several methods to choose theoptimal rules: Γ-minimax rules, conditional Γ-minimax rules, the most stable rules andthe posterior regret Γ-minimax (PRGM) rules.In the traditional insurance premium calculation, people often use the symmetricloss functions to estimate the policy holder’s risk, such as the square error loss functionthe absolute loss function and so on. The symmetric loss function in certain situations issuitable, and is also relatively easy calculation; what’s more, it assigns equal penalty toover-and underestimation. But in some estimation problems, it may be inappropriate touse the symmetric loss functions; the penalty for overestimation may not have the sameas for underestimation. At this time, one has to consider asymmetric loss functions. Inthis thesis, we consider two asymmetric loss functions: the Entropy loss function andthe LINEX loss function, to calculate the Bayesian premium.In this paper, we choose five classes of priors, in the Poisson-Gamma model,obtain Bayesian premiums by using posterior regret Γ-minimax methodology.Symmetric loss function (square error loss) and asymmetric loss function (Entropy loss,LINEX loss) are considered. Using numerical example, we compare the posteriorregret Γ-minimax premium under Γ-contaminated classes and assess the effect, andalso compare the risk of three loss functions, the result show that asymmetric lossfunction can measure risk better and obtain premium fairer than symmetric lossfunction.