As one of the earliest crucial applications of Bayesian statistics, credibility theory (see Bühlmann and Gisler, 2006) was first developed for net premium calibration in insurance by optimally combining individual claim history with other claim histories from the whole population; for a century, this research direction has become a major discipline in the interplay among actuarial science, operational research and statistics. Traditionally, for the ease of calibration, the credibility formula is a linear functional of historical observations, which greatly simplifies the underlying computational complexity; yet, its downside is the resulting high sensitivity towards outliers. To remedy this shortcoming, De Vylder (1976) proposed to first transform the observations collected by truncation in particular, and this semi-linear approach was further investigated in Bühlmann & Gisler (2006). Gisler (1980) suggested that the $L^2$-optimal truncation point can be determined in an ad hoc manner, but the derivation of its general explicit formula is difficult. In our present work, to strike a balance between practical usage and mathematical tractability, we focus on heterogeneous risks all coming from possibly different maximum domain of attractions of the extreme value distributions, which well suffices in practice. By incorporating the satisficing method commonly used in operational research, we close the gap by providing the explicit formula for the aforementioned optimal truncation point up to a slowly varying function of the sample size in an asymptotic sense. A comprehensive numerical study also illuminates that with the aid of this newly obtained truncation point, the corresponding semi-linear credibility formula outperforms the classical Bühlmann model.

作为贝叶斯统计最早的重要应用之一，可信度理论（参见 Bühlmann 和 Gisler，2006 年）最初是通过将个人索赔历史与整个人口的其他索赔历史最佳结合来校准保险的净保费；一个世纪以来，这一研究方向已成为精算学、运筹学和统计学相互作用的主要学科。传统上，为了便于标定，可信度公式是历史观测值的线性函数，大大简化了底层的计算复杂度；然而，它的缺点是对异常值的高度敏感性。为了弥补这个缺点，De Vylder (1976) 提出首先对截断收集的观测值进行变换，特别是，Bühlmann & Gisler (2006) 进一步研究了这种半线性方法。Gisler (1980) 认为${升}^2$- 最佳截断点可以通过临时方式确定，但其一般显式公式的推导很困难。在我们目前的工作中，为了在实际使用和数学易处理性之间取得平衡，我们关注来自极值分布的可能不同最大吸引力域的异质风险，这在实践中就足够了。通过结合运筹学中常用的令人满意的方法，我们通过提供上述最佳截断点的明确公式来缩小差距，直到渐近意义上的样本大小的缓慢变化函数。综合数值研究还表明，借助这个新获得的截断点，相应的半线性可信度公式优于经典的 Bühlmann 模型。