This paper exploits the representation of the conditional mean risk sharing allocations in terms of size-biased transforms to derive effective approximations within insurance pools of limited size. Precisely, the probability density functions involved in this representation are expanded with respect to the Gamma density and its associated Laguerre orthonormal polynomials, or with respect to the Normal density and its associated Hermite polynomials when the size of the pool gets larger. Depending on the thickness of the tails of the loss distributions, the latter may be replaced with their Esscher transform (or exponential tilting) of negative order. The numerical method then consists in truncating the series expansions to a limited number of terms. This results in an approximation in terms of the first moments of the individual loss distributions. Compound Panjer-Katz sums are considered as an application. The proposed method is compared with the well-established Panjer recursive algorithm. It appears to provide the analyst with reliable approximations that can be used to tune system parameters, before performing exact calculations.

本文利用条件平均风险分担分配在规模偏向变换方面的表示，在有限规模的保险池中推导出有效的近似值。准确地说，当池的大小变大时，此表示中涉及的概率密度函数相对于 Gamma 密度及其相关的 Laguerre 正交多项式，或相对于正态密度及其相关的 Hermite 多项式进行了扩展。根据损失分布尾部的厚度，后者可以用负序的埃舍尔变换（或指数倾斜）代替。然后数值方法包括将级数展开截断为有限数量的项。这导致在各个损失分布的一阶矩方面的近似值。复合 Panjer-Katz 和被视为一个应用程序。将所提出的方法与完善的 Panjer 递归算法进行比较。它似乎为分析人员提供了可靠的近似值，可用于在执行精确计算之前调整系统参数。