  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.
