The primary objective of this scholarly work is to develop two estimation procedures – maximum likelihood estimator (MLE) and method of trimmed moments (MTM) – for the mean and variance of lognormal insurance payment severity data sets affected by different loss control mechanism, for example, truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance proportions), in insurance and financial industries. Maximum likelihood estimating equations for both payment-per-payment and payment-per-loss data sets are derived which can be solved readily by any existing iterative numerical methods. The asymptotic distributions of those estimators are established via Fisher information matrices. Further, with a goal of balancing efficiency and robustness and to remove point masses at certain data points, we develop a dynamic MTM estimation procedures for lognormal claim severity models for the above-mentioned transformed data scenarios. The asymptotic distributional properties and the comparison with the corresponding MLEs of those MTM estimators are established along with extensive simulation studies. Purely for illustrative purpose, numerical examples for 1500 US indemnity losses are provided which illustrate the practical performance of the established results in this paper.

中文翻译：

---

对数正态保险支付严重性数据损失模型的稳健估计

这项学术工作的主要目标是开发两种估计程序——最大似然估计（MLE）和修剪时刻的方法(MTM) – 对于受不同损失控制机制影响的对数正态保险支付严重性数据集的均值和方差，例如，截断（由于免赔额）、审查（由于政策限制）和缩放（由于共同保险比例），在保险和金融行业。导出了每次付款和每次损失付款数据集的最大似然估计方程，可以通过任何现有的迭代数值方法轻松求解。这些估计量的渐近分布是通过 Fisher 信息矩阵建立的。此外，为了平衡效率和稳健性并去除某些数据点的点质量，我们为上述转换数据场景的对数正态索赔严重性模型开发了动态 MTM 估计程序。渐近分布特性以及与这些 MTM 估计器的相应 MLE 的比较是与广泛的模拟研究一起建立的。仅出于说明目的，提供了 1500 次美国赔偿损失的数值示例，以说明本文中既​​定结果的实际性能。