Abstract
Actuaries are interested in modeling actuarial data using loss models that can be adopted to describe risk exposure. This paper introduces a new flexible extension of the log-logistic distribution, called the extended log-logistic (Ex-LL) distribution, to model heavy-tailed insurance losses data. The Ex-LL hazard function exhibits an upside-down bathtub shape, an increasing shape, a J shape, a decreasing shape, and a reversed-J shape. We derived five important risk measures based on the Ex-LL distribution. The Ex-LL parameters were estimated using different estimation methods, and their performances were assessed using simulation results. Finally, the performance of the Ex-LL distribution was explored using two types of real data from the engineering and insurance sciences. The analyzed data illustrated that the Ex-LL distribution provided an adequate fit compared to other competing distributions such as the log-logistic, alpha-power log-logistic, transmuted log-logistic, generalized log-logistic, Marshall–Olkin log-logistic, inverse log-logistic, and Weibull generalized log-logistic distributions.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
We introduce a flexible extension of the log-logistic distribution called the extended log-logistic (Ex-LL) distribution
Some risk measures are obtained for the Ex-LL. Distribution along with their detailed simulation results which illustrate that the tail of the Ex-LL distribution is heavier than the tails of the log-logistic and exponentiated loglogistic distributions
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Modeling insurance losses data has received significant interest from actuaries and risk managers who often evaluate and study the unlikely outcomes that the value-at-risk may express by chance. The insurance data are usually unimodal [1], right-skewed [2], positive [3], have a heavy-tailed density [4], and have a unimodal hump shape [5]. There is a clear need to develop and propose more flexible distributions by extending the well-known classical distributions or by introducing a new family to model several insurance datasets such as financial returns, unemployment insurance data, insurance losses data, and risk management data, among others
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