Truncated, Censored, and Actuarial Payment–type Moments for Robust Fitting of a Single-parameter Pareto Distribution

With some regularity conditions maximum likelihood estimators (MLEs) al-ways produce asymptotically optimal (in the sense of consistency, efficiency, sufficiency,and unbiasedness) estimators. But in general, the MLEs lead to non-robust statisticalinference, for example, pricing models and risk measures. Actuarial claim severity iscontinuous, right-skewed, and frequently heavy-tailed. The data sets that such mod-els are usually fitted to contain outliers that are difficult to identify and separate fromgenuine data. Moreover, due to commonly used actuarial “loss control strategies” infinancial and insurance industries, the random variables we observe and wish to modelare affected by truncation (due to deductibles), censoring (due to policy limits), scal-ing (due to coinsurance proportions) and other transformations. To alleviate the lackof robustness of MLE-based inference in risk modeling, here in this paper, we proposeand develop a new method of estimation –method of truncated moments(MTuM) andgeneralize it for different scenarios of loss control mechanism. Various asymptotic prop-erties of those estimates are established by using central limit theory. New connectionsbetween different estimators are found. A comparative study of newly-designed methodswith the corresponding MLEs is performed. Detail investigation has been done for asingle parameter Pareto loss model including a simulation study