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Abstract
Generalized linear models might not be appropriate when the probability of extreme events is higher than that implied by the normal distribution. Extending the method for estimating the parameters of a double Pareto lognormal distribution (DPLN) in Reed and Jorgensen (2004), we develop an EM algorithm for the heavy-tailed Double-Pareto-lognormal generalized linear model. The DPLN distribution is obtained as a mixture of a lognormal distribution with a double Pareto distribution. In this paper the associated generalized linear model has the location parameter equal to a linear predictor which is used to model insurance claim amounts for various data sets. The performance is compared with those of the generalized beta (of the second kind) and lognorma distributions.
Highlights
Heavy-tailed distributions are an important tool for actuaries working in insurance where many insurable events have low likelihoods and high severities and the associated insurance policies require adequate pricing and reserving
We propose in this article the use of the double Pareto lognormal (DPLN) distribution as an alternative model for heavy-tailed events
In respect of model selection, we provide the negative of the maximum of the log-likelihood (NLL), Akaike’s information criterion (AIC) and Bayesian information criterion (BIC) results in the table
Summary
Heavy-tailed distributions are an important tool for actuaries working in insurance where many insurable events have low likelihoods and high severities and the associated insurance policies require adequate pricing and reserving. In such cases the four-parameter generalized beta distribution of the second kind (GB2) and the three-parameter generalized gamma distribution fulfil this purpose, as demonstrated in McDonald (1990), Wills et al (2006), Frees and Valdez (2008), Wills et al (2006), Frees et al (2014a) and Chapter 9 of Frees et al (2014b). Particular applications of the DPLN distribution to insurance and actuarial science have previously been given in Ramírez-Cobo et al (2010) and Hürlimann (2014)
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