Abstract
We construct a nonparametric sequential test for the ruin probability and a corresponding change-point test in a risk model perturbed by diffusion. Some limiting properties are derived, which extend and improve on recent results of Conti (Stat Prob Lett 72:333–343, 2005) and Jahnke (Diploma thesis, University of Cologne, 2007). It is shown that the monitoring procedures can be designed such that the tests have an asymptotic prescribed false alarm rate (size) α and power 1. Some results from a small simulation study are also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andersen ES (1957) On the collective theory of risk in the case of contagion between the claims. Trans XVth Int Congr Actuar II(6): 219–229
Aue A, Horváth L (2004) Delay time in sequential detection of change. Stat Prob Lett 67: 221–231
Chu C-SJ, Stinchcombe M, White H (1996) Monitoring structural change. Econometrica 64: 1045–1065
Conti PL (2005) A nonparametric sequential test with power 1 for the ruin probability in some risk models. Stat Prob Lett 72: 333–343
Csörgő M, Steinebach J (1991) On the estimation of the adjustment coefficient in risk theory via intermediate order statistics. Insur Math Econ 10: 37–50
Csörgő S (1982) The empirical moment generating function. In: Gnedenko BV, Puri ML, Vincze I (eds) Colloquia Mathematica Societatis János Bolyai, vol 32. Nonparametric Statistical Inference. Elsevier, Amsterdam, pp 139–150
Csörgő S, Teugels JL (1990) Empirical Laplace transform and approximation of compound distributions. J Appl Prob 27: 88–101
Dufresne F, Gerber HU (1991) Risk theory for the compound Poisson process that is perturbed by diffusion. Insur Math Econ 10: 51–59
Feuerverger A (1989) On the empirical saddlepoint approximation. Biometrika 76: 457–464
Furrer HJ, Schmidli H (1994) Exponential inequalites for ruin probabilities of risk processes perturbed by diffusion. Insur Math Econ 15: 23–36
Gerber HU (1970) An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuarietidskrift, pp 205–210
Grandell J (1991) Aspects of Risk Theory. Springer, New York
Horváth L, Hušková M, Kokoszka P, Steinebach J (2004) Monitoring changes in linear models. J Stat Plan Inf 126: 225–251
Jahnke D (2007) Sequentielles Testen von Ruinwahrscheinlichkeiten im Sparre Andersen’schen Risikomodell. Diploma thesis, University of Cologne
Komlós J, Major P, Tusnády G (1975) An approximation of partial sums of independent R.V.’s and the sample D.F.-I. Z Wahrscheinlichkeitstheorie verwandte Geb 32: 111–131
Komlós J, Major P, Tusnády G (1976) An approximation of partial sums of independent R.V.’s and the sample D.F.-II. Z Wahrscheinlichkeitstheorie verwandte Geb 34: 33–58
Mammitzsch V (1986) A note on the adjustment coefficient in ruin theory. Insur Math Econ 5: 147–149
Müller A (1996) Schätzung von Anpassungskoeffizienten in Risikomodellen mit Diffusionskomponenten. Diploma thesis, University of Marburg
Pitts SM, Grübel R, Embrechts P (1996) Confidence bounds for the adjustment coefficient. Adv Appl Prob 28: 802–827
Schmidli H (1995) Cramér–Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Insur Math Econ 16: 135–149
Strassen V (1967) Almost sure behaviour of sums of independent random variables and martingales. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol 2. University of California Press, Los Angeles, pp 315–343
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Steinebach, J.G. Monitoring risk in a ruin model perturbed by diffusion. Metrika 70, 205–224 (2009). https://doi.org/10.1007/s00184-008-0187-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-008-0187-2