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Monotonicity properties and the deficit at ruin in the Sparre Andersen model

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  • Georgios Psarrakos
  • Konstadinos Politis

Abstract

Let Hu(y) be the (proper) distribution function of the deficit at ruin, given that ruin occurs with initial surplus u, in the Sparre Andersen model of risk theory. Dickson & dos Reis (1996) discussed the monotonicity of Hu(y) as a function of u. In this paper, we obtain various monotonicity results for Hu(y) and other related quantities for the decreasing/increasing failure rate (DFR/IFR) and the increasing/decreasing mean residual lifetime (IMRL/DMRL) classes of distributions. These results in particular extend and make more concrete the results of Dickson & dos Reis (1996) and Willmot & Lin (1998). A new class of distributions (increasing convolution ratio; ICR) is introduced. This class extends the well-known class of distributions with IFR. Specifically, we show that if the ladder height distribution F in the model is ICR, the ratio is a non-decreasing function of u, where ψ(u) denotes the ruin probability and . Further, we obtain generalizations (expressed in terms of the distribution of the deficit) of the well-known new worse than used (NWU) property of the probability of non-ruin.

Suggested Citation

  • Georgios Psarrakos & Konstadinos Politis, 2009. "Monotonicity properties and the deficit at ruin in the Sparre Andersen model," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2009(2), pages 104-118.
    • Handle: RePEc:taf:sactxx:v:2009:y:2009:i:2:p:104-118
    DOI: 10.1080/03461230802022169
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