With some regularity conditions maximum likelihood estimators (MLEs) always produce asymptotically optimal (in the sense of consistency, efficiency, sufficiency, and unbiasedness) estimators. But in general, the MLEs lead to non-robust statistical inference, for example, pricing models and risk measures. Actuarial claim severity is continuous, right-skewed, and frequently heavy-tailed. The data sets that such models are usually fitted to contain outliers that are difficult to identify and separate from genuine data. Moreover, due to commonly used actuarial "loss control strategies" in financial and insurance industries, the random variables we observe and wish to model are affected by truncation (due to deductibles), censoring (due to policy limits), scaling (due to coinsurance proportions) and other transformations. To alleviate the lack of robustness of MLE-based inference in risk modeling, here in this paper, we propose and develop a new method of estimation - method of truncated moments (MTuM) and generalize it for different scenarios of loss control mechanism. Various asymptotic properties of those estimates are established by using central limit theory. New connections between different estimators are found. A comparative study of newly-designed methods with the corresponding MLEs is performed. Detail investigation has been done for a single parameter Pareto loss model including a simulation study.

翻译：在满足一定正则性条件下，极大似然估计量（MLEs）总能产生渐近最优（在一致性、有效性、充分性及无偏性意义上）的估计量。但一般而言，极大似然估计量会导致非稳健的统计推断，例如在定价模型和风险度量中的应用。精算索赔严重度具有连续性、右偏性且常呈现厚尾特征。此类模型所拟合的数据集中常包含难以识别并与真实数据分离的异常值。此外，由于金融保险行业中普遍采用的精算"损失控制策略"，我们观察并需要建模的随机变量会受到截断（因免赔额）、删失（因保单限额）、缩放（因共保比例）及其他变换的影响。为解决基于极大似然估计的风险建模推断缺乏稳健性问题，本文提出并发展了一种新的估计方法——截断矩法（MTuM），并将其推广至损失控制机制的不同场景。利用中心极限理论建立了这些估计量的多种渐近性质，发现了不同估计量之间的新关联。将新设计的方法与相应的极大似然估计量进行了比较研究，针对单参数帕累托损失模型（包含模拟研究）开展了详细探讨。