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Abstract

In actuary, the derivation of loss distributions from insurance data is of great interest. Fitting an adequate distribution to real insurance data is not an easy task, mainly due to the nature of the data, which shows several features to be accounted for. Although, because of its stochastic and numerical simplicity, it is often assumed that the involved financial risk factors are normally distributed, but empirical studies indicate that most of financial risk factors have distributions with high peaks and heavy tails. Thus, it is important in the actuarial science to model insurance risks with skewed distributions. Claims size data in non-life insurance policies are very skewed and exhibit high kurtosis and extreme tails. Skew distributions are reasonable models for describing claims in property-liability insurance. We fit several well-known skew distributions (skew-normal, skew-Laplace, generalized logistic, generalized hyperbolic, variance gamma, normal inverse Gaussian, Marshal-Olkin Log-Logistic and Kumaraswamy Marshal-Olkin Log-Logistic distributions) to the amount of automobile accident claims for property damage to a third party. The data are from financial records of a state-owned major general insurance company in Iran. The fitted models are compared using AIC (Akaike information criterion), BIC (Bayesian information criterion) and Kolmogorov-Smirnov goodness-of-fit test statistics. We find that the Kumarasamy Marshal-Olkin Log-Logistic distribution is better than other considered distributions in describing the features of the observed data. This distribution is a very perfect distribution to describe the skew data. The value at risk and conditional tail expectation, as most common risk measures in insurance, are estimated for the data under consideration.

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