损失额分布的非参数估计方法研究

Non-parametric Estimation of Loss Distribution

在财险实务中,理赔额的分布直接影响到了费率厘定,准备金评估,以及"偿二代"下风险资本计算等精算问题上。理赔额是基于损失额确定的,但是损失额的真实分布未知,如果对其进行假设,则分析结果的误差会比较大。本文提出了损失额分布的非参数估计方法,即最大惩罚似然估计法,该方法对损失额分布不作任何假设,损失额分布完全由记录在案的理赔数据决定。新方法引入一种迭代算法来解决约束最优化问题,得到损失额的危险函数和生存函数最大惩罚似然估计,估计曲线平滑,便于分析风险变化的趋势。同时本文也建立了计算估计渐进方差的数学公式,该公式可以用来建立预测值的置信区间。随机模拟结果显示保单数量越大,分布估计越接近于真实值,渐进方差计算公式越精确。在"偿二代"监管框架下,新方法可以被保险公司作为内部模型来进行风险分析。

In general insurance practices,distribution of claim amount has a direct bearing on premium rating,reserve calculation,risk capital evaluation under C-Rosss and so forth. The claim amount is determined on the basis of the loss amount. However,the actual loss distribution is unknown. Putting assumptions on the loss distribution causes bias in calculation. In this paper,a new method,named as maximum penalized likelihood estimation,was proposed to estimate the loss distribution non-parametrically. This proposed method made no assumption on the loss distribution,and the distribution was completely determined by the claim data on record. To obtain estimates of hazard and survival functions for loss amounts,the constrained optimization problem was solved by using an iterative algorithm. The smoothed estimates could facilitate the analysis of the trend of insurance risk changes. The asymptotic variance formula was also derived for the estimators and could be used to construct confidence intervals for predictions. Results from simulation studies showed that,with larger insurance policy size,the estimates were closer to true distributions and the asymptotic variance formula was more accurate. Under C-ROSS framework,the proposed method can be used as an internal model for risk measurement.