Modeling loss data with composite models / Mastoureh Maghsoudi - Mastoureh , Maghsoudi (2016)
Modeling insurance loss data of a unimodal type with a heavy tail is an interesting topic for actuaries. Recently, the focus for modeling such data has been directed towards composite models following a study by Cooray and Ananda (2005). These models are designed by piecing together two weighted distributions at a certain threshold q. For simplicity, the distribution up to the threshold q and the one beyond it are referred to as a head and a tail of the composite model, respectively. Studies have shown that composite models offer some solutions to the problem for covering the whole data sets (Nadarajah and Bakar, 2012). This study proposed a new approach of finding parameters in terms of mixing weight f and threshold q parameters. The advantages of this new procedure is that it has a general formula and can be used for all distributions as a head and a tail in constructing composite models. This leads to the composite model being more flexible than the previous models. In previous studies, all parameters in the composite model were estimated in terms of two parameters which are the mixing weight and one of the head distribution parameters. New composite models with a tail distribution that belongs to the transformed Beta, transformed Gamma, and inverse transformed Gamma families are introduced and fitted to two well-known data sets in actuarial industries. Based on the negative log-likelihood, (NLL), Akaike information criteria, (AIC), and Schwarz’s Bayesian criterion, (SBC), the compositeWeibull- inverse Paralogistic model and the compositeWeibull-inverse transformed Gamma model are found (with the lowest value in all above goodness-of-fit measurements) to have the best fit among all composite models considered. For all new composite models that have been introduced, mixing weight f, threshold q, and moment generating function are obtained and for the two best fitted composite models, more statistical properties such as mean, variance, skewness, and kurtosis are calculated. In this study, two real insurance loss data sets are used to illustrate the performance of these new composite models. The first data set is Danish fire insurance loss data which comprises of 2492 losses arising from Copenhagen. The second data set is the allocated loss adjustment expenses (ALAE) data set which consists of 1500 general liability losses recorded in US dollars. This study also presents the applications of the fitted composite models in risk measurements. Two well-known risk measures are employed, namely, the value-at-risk (VaR) and the conditional tail expectation (CTE) for all composite models in transformed Beta, transformed Gamma, and inverse transformed Gamma families. To test the validity of the value-at-risk and the conditional tail expectation, backtesting method is applied. All the computations includes constructing the composite model, fitting the distributions and composite models, graphing, measuring the risk measures, and backtesting were performed using the statistical software R.