Modeling with Weibull-Pareto Models

Abstract

In this paper we develop several composite Weibull-Pareto models and suggest their use to model loss payments and other forms of actuarial data. These models all comprise a Weibull distribution up to a threshold point, and some form of Pareto distribution thereafter. They are similar in spirit to some composite lognormal-Pareto models that have previously been considered in the literature. All of these models are applied, and their performance compared, in the context of a realworld fire insurance data set.

A Note on Parameter Estimation in the Composite Weibull–Pareto Distribution

“…The Weibull distribution (W), Pareto type II or Lomax distribution (L) and the CWL distribution have been firstly fitted to Danish fire insurance losses using the functions mle and mle2 in R. The results coincide with those reported in Table 1 in Scollnik and Sun (2012) and in Table 2 in Bakar et al (2015), obtained using the nlm optimization function in R. For the CWL model, the estimated value for the threshold θ was 0.9717. The corresponding standard error was 0.0069.…”

“…Clearly, conditions (i) and (ii) are satisfied, as they are imposed in the third composite Weibull-Pareto (CWL) model in Scollnik and Sun (2012); it is simple to see that condition (iii) implies that φ =…”

“…They proposed models with the tail belonging to a family of transformed beta distributions. One of the models discussed in that paper, the Weibull-Pareto, was previously studied by Cooray (2009) and also by Scollnik and Sun (2012). Bakar et al (2015) define the probability density function (PDF) of the latter composite model in the following way:…”

“…This model was fitted by Scollnik and Sun (2012) and Bakar et al (2015) to a dataset that consisted of 2492 fire losses, adjusted for inflation, arising from claims in Copenhagen. It was recorded in millions of Danish krone (DKK) for a period from 1980 to 1990.…”

Composite Weibull-Inverse Transformed Gamma distribution and its actuarial application

“…Pigeon and Denuit [8] mentioned a heterogeneity of the threshold parameter situation in a composite lognormal-Pareto model that can be vary among observations. Scollnik and Sun [9] distribution is related to Nadarajah and Bakar [10] the aim at improving the fit to the Danish data fitting. They introduced the new model which is called composite lognormal-Burr model.…”