Fitting mixtures of Erlangs to censored and truncated data using the EM algorithm

Modeling data on claim sizes is crucial when pricing insurance products. Such loss models require on the one hand the flexibility of nonparametric density estimation techniques to describe the insurance losses and on the other hand the feasibility to analytically quantify the risk. Mixtures of Erlang distributions with a common scale are very versatile as they are dense in the space of distributions on ℝ+ (Tijms (1994, p. 163)). At the same time, it is possible to work analytically with this kind of distributions. Closed-form expressions of quantities of interest, such as the Value-at-Risk (VaR) and the Tail-Value-at-Risk (TVaR), can be derived as well as appealing closure properties (Lee and Lin (2010), Willmot and Lin (2011) and Klugman et al. (2012)). In particular, using these distributions in aggregate loss models leads to an analytical form of the corresponding aggregate loss distribution which avoids the need for simulations to evaluate the model. In actuarial science, claim severity data is often censored and/or truncated due to policy modifications such as deductibles and policy limits. Lee and Lin (2010) formulate a calibration technique based on the EM algorithm for fitting mixtures of Erlangs with a common scale parameter to complete data. Here, we construct an adjusted EM algorithm which is able to deal with censored and truncated data, inspired by McLachlan and Peel (2001) and Lee and Scott (2012). Using the developed R program, we demonstrate the strength of mixtures of Erlangs to approximate skew, bimodal, moderately heavy tailed or even flat densities based on simulated censored and truncated samples.

Jaar van publicatie:2013

BOF-keylabel:ja

IOF-keylabel:ja

BOF-publication weight:1

CSS-citation score:2

Auteurs:International

Authors from:Higher Education

Toegankelijkheid:Open