截断与删失对数正态分布稳健拟合的Winsorized矩方法
Method of Winsorized Moments for Robust Fitting of Truncated and Censored Lognormal Distributions
AI总结：
本文提出自适应Winsorized矩方法（MWM）用于截断与删失对数正态分布的稳健参数估计，推导其渐近性质并与MLE和MTM比较，通过模拟和真实保险损失数据验证了该方法的有效性。
摘要：
在构建参数模型以预测未来索赔成本时，必须考虑几个重要细节：（i）模型设计应能适应免赔额、保单限额和共保因子；（ii）参数估计应具有稳健性，以控制离群值对模型预测的影响；（iii）所有点预测都应辅以不确定性估计。本文提出的方法为同时解决上述所有方面提供了一个框架。利用每次赔付（payment-per-payment）和每次损失（payment-per-loss）变量，我们构建了截断与删失对数正态分布参数的自适应Winsorized矩方法（MWM）估计量。进一步推导了该方法的渐近分布性质，并将其与最大似然估计量（MLE）和截尾矩方法（MTM）估计量进行比较，后者是MWM的主要竞争对手。此外，通过广泛的模拟研究和风险度量敏感性分析验证了理论结果。最后，使用被广泛研究的1500个美国赔偿损失数据集说明了这些方法的实际性能。利用这一真实数据集还证明，与单独拟合的分布相比，复合模型在预测模型质量方面并未带来显著改进，尤其是对于截断和删失样本数据而言。
英文摘要：
When constructing parametric models to predict the cost of future claims, several important details have to be taken into account: (i) models should be designed to accommodate deductibles, policy limits, and coinsurance factors, (ii) parameters should be estimated robustly to control the influence of outliers on model predictions, and (iii) all point predictions should be augmented with estimates of their uncertainty. The methodology proposed in this paper provides a framework for addressing all these aspects simultaneously. Using payment-per-payment and payment-per-loss variables, we construct the adaptive version of method of winsorized moments (MWM) estimators for the parameters of truncated and censored lognormal distribution. Further, the asymptotic distributional properties of this approach are derived and compared with those of the maximum likelihood estimator (MLE) and method of trimmed moments (MTM) estimators. The latter being a primary competitor to MWM. Moreover, the theoretical results are validated with extensive simulation studies and risk measure sensitivity analysis. Finally, practical performance of these methods is illustrated using the well-studied data set of 1500 U.S. indemnity losses. With this real data set, it is also demonstrated that the composite models do not provide much improvement in the quality of predictive models compared to a stand-alone fitted distribution specially for truncated and censored sample data.
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