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Modelling actuarial data with a composite lognormal-Pareto model. (English) Zbl 1143.91027

The Pareto model covers the behaviour of large losses well, but fails to cover the behaviour of small losses. The lognormal model covers the behaviour of small losses well, but fails to cover the behaviour of large losses. In order to achieve both of these behaviours in one model the authors look for a desirable composite model, which took the two-parameter lognormal density up to an unknown threshold value and the two-parameter Pareto density for the rest of the model. The proposed composite density can be reparameterized and re-written as \[ f(x)=\begin{cases} {{\alpha\theta^{\alpha}}\over{(1+\Phi(k))x^{\alpha+1}}} \exp\left\{-{\alpha^{2}\over2k^2}\ln^{2}(x/\theta)\right\}& \text{if \(0<x\leq\theta\)}\\ {\alpha\theta^{\alpha}\over(1+\Phi(k))x^{\alpha+1}}& \text{if \(\theta\leq x<\infty\)},\end{cases} \] where \(\Phi(\cdot)\) is the cumulative distribution function of the standard normal distribution and \(k\) is a known constant which is given by the positive solution of the equation \(\exp(-k^2)=2\pi k^2\). Properties and behaviour of composite lognormal-Pareto model are discussed. The authors study the maximum likelihood parameter estimation methods and related techniques for complete and censored data to the composite lognormal-Pareto model.

MSC:

91B30 Risk theory, insurance (MSC2010)
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