Abstract
In this article we introduce the stability analysis of a compound sum: it consists of computing the standardized variation of the survival function of the sum resulting from an infinitesimal perturbation of the common distribution of the summands. Stability analysis is complementary to the classical sensitivity analysis, which consists of computing the derivative of an important indicator of the model, with respect to a model parameter. We obtain a computational formula for this stability from the saddlepoint approximation. We apply the formula to the compound Poisson insurer loss with gamma individual claim amounts and to the compound geometric loss with Weibull individual claim amounts.
Highlights
This article presents a computational formula for the stability of the survival function (s.f.) of the compound sum of independent and identically distributed (i.i.d.) random variables that are independent of their summation index
We define the stability of a sum as the standardized variation of the s.f. of the sum resulting from an infinitesimal perturbation at some point x ∈ R of the distribution of the summands
Their applications concerned statistical testing. They computed the tail area influence function with the saddlepoint approximation of Daniels (1954). This article generalizes this approximation to the stability of the compound sum and suggests using this concept in risk management
Summary
This article presents a computational formula for the stability of the survival function (s.f.) of the compound sum of independent and identically distributed (i.i.d.) random variables that are independent of their summation index. Fxε with respect to (w.r.t.) ε evaluated at ε = 0 is the s.f. stability (s.f.s.) at the perturbation point x This concept differs from the one of sensitivity of queueing theory or risk theory, which is defined as the derivative of the s.f. of the sum w.r.t. a parameter of F (cf., e.g., Asmussen and Albrecher (2010), sct. For a given actuarial aggregate loss model in the form of a compound sum, if a stability of low magnitude results from the perturbation in the form of a new large individual claim amount They computed the tail area influence function with the saddlepoint approximation of Daniels (1954) This article generalizes this approximation to the stability of the compound sum and suggests using this concept in risk management. Some related long derivatives are provided in the Appendix A
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