Robust Estimation of Parametric Models for Insurance Loss Data

Abstract

Parametric statistical models for insurance claims severity are 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,” 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. In the current practice, statistical inference for loss models is almost exclusively likelihood (MLE) based, which typically results in non-robust parameter estimators, pricing models, and risk measures. To alleviate the lack of robustness of MLE-based inference in risk modeling, two broad classes of parameter estimators - Method of Trimmed Moments (MTM) and Method of Winsorized Moments (MWM) - have been recently developed. MTM and MWM estimators are sufficiently general and flexible, and possess excellent large- and small- sample properties, but they were designed for complete (not transformed) data. In this dissertation, we first redesign MTM estimators to be applicable to claim severity models that are fitted to truncated, censored, and insurance payments data. Asymptotic properties of such estimators are thoroughly investigated and their practical performance is illustrated using Norwegian fire claims data. In addition, we explore several extensions of MTM and MWM estimators for complete data. In particular, we introduce truncated, censored, and insurance payment-type estimators and study their asymptotic properties. Our analysis establishes new connections between data truncation, trimming, and censoring which paves the way for more effective modeling of non-linearly transformed loss data.