Modeling with the composite lognormal-Pareto models and the composite Weibull-Pareto models
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2010
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Abstract
The lognormal and Pareto distributions are frequently applied in the actuarial and insurance industries to model their loss payment data. This data is often highly positively skewed. The Pareto distribution is often used to model the larger data values since it has a longer and thicker upper tail. On the other hand, the lognormal distribution is often appropriate for the larger data values with lower frequencies and smaller data values with higher frequencies. Even though the lognormal model covers larger data with lower frequencies, it fades away to zero more quickly than the Pareto model. Furthermore, the Pareto model does not provide a reasonable parametric fit for smaller data due to the monotonic decreasing shape of the density. Therefore, taking into account the tail behavior of both small and large losses, Cooray and Ananda (2005) proposed a two-parameter lognormal density up to an unknown threshold value and a two-parameter Pareto density thereafter. The resulting two-parameter smooth density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behaviour is quite similar to the Pareto density. Scollnik (2007) analyzed limitations of the composite lognormal-Pareto model which are likely to severely curtail its potential for practical application to real world data sets and presented two different extended models. Parameter estimation techniques and properties of this new composite lognormal-Pareto model are discussed and we compare its performance with the other commonly used models. Both the maximum likelihood estimation and Bayesian estimation methods are applied to compare the performance of these composite models. A simulat d example, a well-known Danish fire insuranc data set, and a Guinea pig survival time data set, together with a group of arm A head and neck cancer data, are all analyzed to show the applicability of the three composite lognormal-Pareto models as well as the Weibull- Pareto composit model proposed by Preda and Ciumara (2006) , and further examined by Cooray(2009). Similarly, we provide two extensions of the composite Weibull-Pareto model to fix the problem of the composite Weibull-Pareto model (Preda and Ciumara, 2006).
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Bibliography: p. 76-79
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Sun, C. (2010). Modeling with the composite lognormal-Pareto models and the composite Weibull-Pareto models (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/3874