Burr Distribution as an Actuarial Risk Model and the Computation of Some of Its Actuarial Quantities Related to the Probability of Ruin

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.

Modeling of Insurance Data through Two Heavy Tailed Distributions: Computations of Some of Their Actuarial Quantities through Simulation from Their Equilibrium Distributions and the Use of Their Convolutions

“…In both the cases, the Probability of ultimate ruin was found to be a decreasing function of the initial surpluses and this is expected, as larger initial surpluses should diminish the chance of ruin (if any) for the insurance company. The probability of ultimate ruin for the Burr XII distribution was computed through two numerical algorithms namely the stable recursive algorithm and the method of product integration in [13]. It is observed that there is deviation in the values obtained through these two algorithms and the Pollaczek-Khinchin formula.…”

“…3) The simulation schemes need to be further improved for the values of the probability of ultimate ruin for the Burr XII distribution, it yielded are inconsistent to the values computed earlier [13]. 4) To avoid the complexity of having to evaluate a number of nested integrals numerically, we had to remain content with the evaluation of just the lower order convolution of these distributions, which, in turn, enabled us to compute the probability function for the number of claims until ruin, for a lower ranking whose significance to reality is not as important as the function, computed for a higher ranking of the order at which the claims arrive.…”

“…If we consider a sample of "m" items 1 2 , , , m d d d  from the Weibull distribution whose pdf is given by (2.1.2), the log likelihood function is given by We have used the Multi Parameter Newton Raphson Method for estimating the parameters of the Weibull distribution. In the appendix: (1) of [13] we have given a very brief introduction to the Multi Parameter Newton Raphson method and have obtained the gradient and hessian matrices for the Weibull distribution which are required in executing the Multi Parameter Newton Raphson method for estimating the parameters of this distribution.…”

“…In the appendix: (1) of [13], we have obtained the Gradient and the Hessian Matrices for estimating the parameters of the Burr XII distribution through the algorithm stated in [12]. Let u T denote the time to ruin from initial surplus u so that…”

On Prognosis of the Competitiveness of Industrial Enterprises

“…Currently, one of the important examples of markets, which characterizing by the crowding out of some products by others that are more attractive from a technological point of view, is the market for information and telecommunication technologies (the market for data transmission services). For most managers of enterprises that are suppliers of products and investors, the issue of competitiveness of enterprises and their development prospects [1][2][3][4][5][6][7][8][9][10] and its prognosis [11][12][13] is relevant. The competitiveness of an enterprise is determined, first of all, by the quantity of manufactured products and sold, as well as the profit received.…”