Loss modeling with the size-biased lognormal mixture and the entropy regularized EM algorithm

The Erlang mixture with a common scale parameter is one of many popular models for modeling insurance losses. However, the actuarial literature recognizes and discusses some limitations of aforementioned model in approximate heavy-tailed distributions. In this paper, a size-biased left-truncated Lognormal (SB-ltLN) mixture is proposed as a robust alternative to the Erlang mixture for modeling left-truncated insurance losses with a heavy tail. The weak denseness property of the weighted Lognormal mixture is studied along with the tail behavior. Explicit analytical solutions are derived for moments and Tail Value at Risk based on the proposed model. An extension of the regularized expectation–maximization (REM) algorithm with Shannon's entropy weights (ewREM) is introduced for parameter estimation and variability assessment. The Operational Riskdata eXchange's left-truncated internal fraud loss data set is used to illustrate applications of the proposed model. Finally, the results of a simulation study show promising performance of the proposed SB-ltLN mixture in different simulation settings.