Abstract
In this paper, we have used an algorithm to fit the Burr XII distribution to a set of insurance data. As it is well known, the probability of ultimate ruin is obtained as a solution to an integro-differential equation and in case, the claim severity is distributed as Burr XII distribution, this equation has to be solved numerically to obtain an approximation to the probability of ultimate ruin. Two numerical algorithms, namely the stable recursive algorithm and the method of product integration have been used to obtain numerically an approximation to this probability of ultimate ruin. The use of these two numerical algorithms provides a scope for comparing the consistency in values obtained by them. The first two moments of the time to ruin in case of Burr XII distributed claim severity have also been computed using the probability of ultimate ruin obtained through the stable recursive algorithm as an input. All these computations have been done under the assumption of the classical risk model.
Highlights
An actuarial risk model is concerned with the study of the mathematical aspects observed in the behavior of a collection of risks generated by an insurance portfolio
Our main objective was to compute the probability of ultimate ruin in case the claim severity is distributed as Burr XII distribution and this was implemented through the application of two numerical algorithms
Second order moment of the time to ruin for Burr XII with α = 4.21652, τ = 1.2746 and φ = 271225.2
Summary
An actuarial risk model is concerned with the study of the mathematical aspects observed in the behavior of a collection of risks generated by an insurance portfolio. There can be other suitable models for loss modeling in general insurance, we are primarily concerned with the Burr distribution as a loss model for our claim data and have concentrated on the computation of various actuarial quantities like the probability of ruin and the moments of the time to ruin when the loss model or claim severity model is Burr XII. The first part of the paper deals with the Watkins [9] algorithm for obtaining the MLE of the parameters of the Burr XII distribution, followed by testing the goodness of fit through some statistics based on the empirical distribution function (EDF) This is followed by the computation of the probability of ultimate ruin for various values of the initial surplus through the adaptation of the two above mentioned numerical methods. The illustrative Burr XII distribution that is being used is the one that is being fitted to the Property Claim Services (PCS) dataset covering losses resulting from natural catastrophic events in USA that occurred between 1990 and 1999 [10]
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