矩相关保费原理中风险保费的经验厘定

Experience rating of risk premium for moment-related premium principle

在传统的Bühlmann信度理论中,信度估计仅仅适合净保费原理,并且很难直接推广到更一般的保费原理中。本文根据随机变量的矩母函数定义一种统一的保费原理-矩相关保费原理,进而,将信度理论的思想运用于估计风险随机变量的矩母函数,给出矩相关保费原理中风险保费的经验厘定估计,并证明估计的统计性质。结果表明,在净保费原理和指数保费原理中,已有的信度估计是本文估计的特殊情形;在方差保费原理中,本文得到的估计要优于已有的信度估计。最后,通过数值模拟的方法验证新的信度估计的相合性和渐近正态性,并在小样本条件下比较本文估计与已有估计的均方误差。

The traditional experience ratemaking in terms of B¨uhlmann’s credibility theory can only compute experience net premiums and is difficult to be transplanted to general premium calculation principles. This paper unifies a dictionary of moment-related premium principles that can be expressed as functionals of moment generating functions. The credibility idea is applied to moment generating functions and the empirical Bayesian estimates of risk premiums are established. Theoretical properties of this new ratemaking are also discussed. The results show that our premiums are exactly the same as the classical ones under net and exponential principles and are better than that of existing estimates under variance premium principle. We also provide some simulations to examine the consistency and asymptotic normality of the premiums and to compare these newly proposed premiums with existing ones under a few premium principles.