The Log-Gamma Distribution and Non-Normal Error
By Leigh Joseph Halliwell
Because insured losses are positive, loss distributions start from zero and are right-tailed. However, residuals, or errors, are centered about a mean of zero and have both right and left tails. Seldom do error terms from models of insured losses seem normal. Usually they are positively skewed, rather than symmetric. And their right tails, as measured by their asymptotic failure rates, are heavier than that of the normal. As an error distribution suited to actuarial modeling this paper presents and recommends the log-gamma distribution and its linear combi-nations, especially the combination known as the generalized logistic distribution. To serve as an example, a generalized logistic distribution is fitted by maximum likelihood to the standardized residuals of a loss-triangle model. Much theory is required for, and occasioned by, this presentation, most of which appears in three appendices along with some related mathematical history.
Keywords Log-gamma, digamma, logistic, Euler-Mascheroni, cumulant, maximum likelihood, robust, bootstrap