Robust Modelling of Heavy-tailed Insurance Claim Severity Data

Maximum likelihood estimator (MLE) is a widely adopted approach for estimating parameters of claim distributions. While under MLE each observation has the same relative influence on the estimated parameters, correct specification and extrapolation of larger claims are often more important than those of smaller claims in insurance loss modelling and ratemaking perspective. Further, MLE can fail in many practical cases when the estimated tail distribution is heavily distorted by smaller claims. We therefore propose a maximum weighted likelihood estimator (MWLE) which assigns larger weights to larger claim sizes, to mitigate model misspecification risks, resulting in more robust tail estimations. Asymptotic theories are developed to ensure that MWLE is consistent and asymptotically normal, and that model uncertainties can be easily quantified. Also, MWLE can be statistically interpreted as random truncated distributions, so an EM-based algorithm can be applied for efficient parameter estimations. Our approach is exemplified on simulation studies and a Greek automobile insurance dataset.