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.

翻译：本文的主要目标是为保险与金融行业中受不同损失控制机制（例如：免赔额导致的截断、保单限额导致的删失、共保比例导致的缩放）影响的对数正态保险支付严重度数据集，开发两种均值与方差的估计方法——极大似然估计（MLE）与截尾矩估计（MTM）。针对每笔赔付与每笔损失两种支付数据集，推导了可通过现有迭代数值方法直接求解的极大似然估计方程，并通过Fisher信息矩阵建立了这些估计量的渐近分布。进一步地，为兼顾效率与鲁棒性并消除特定数据点的点质量，我们针对上述变换后的数据场景，开发了对数正态索赔严重度模型的动态MTM估计程序。通过理论推导与大范围仿真研究，建立了这些MTM估计量的渐近分布性质及其与对应MLE估计量的比较。为纯说明目的，本文提供了1500例美国赔偿损失的数值算例，展示了所建立结果的实际应用性能。