The Log-Gamma Distribution and Non-Normal Error

By Leigh Joseph Halliwell

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Abstract

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

Citation

Halliwell, Leigh Joseph, "The Log-Gamma Distribution and Non-Normal Error," Variance 13:2, 2021, pp. 173-189.

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Variance (ISSN 1940-6452) is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submissions by non-CAS members is encouraged.