Bounds for compound distributions based on mean residual lifetimes and equilibrium distributions. (English) Zbl 0924.62110

The author establishes bounds for the right tail of the total claims distribution. Upper, lower and moment based Pareto bounds are obtained. Estimates are given in terms of the equilibrium distribution and the mean residual lifetime. Some generalizations with higher-order equilibrium distributions are given.


62P05 Applications of statistics to actuarial sciences and financial mathematics
62E15 Exact distribution theory in statistics
91B30 Risk theory, insurance (MSC2010)
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[1] Abramowitz, M.; Stegun, I., (Handbook of Mathematical Functions (1965), Dover: Dover New York)
[2] Barlow, R.; Proschan, F., Statistical Theory of Reliability and Life Testing: Probability Models (1975), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York · Zbl 0379.62080
[3] Fagiuoli, E.; Pellerey, F., New partial orderings and applications, Naval Research Logistics, 40, 829-842 (1993) · Zbl 0795.62050
[4] Fagiuoli, E.; Pellerey, F., Preservation of certain classes of life distributions under Poisson shock models, Journal of Applied Probability, 31, 458-465 (1994) · Zbl 0806.60075
[5] Gertsbakh, I., Statistical Reliability Theory (1989), Marcel Dekker: Marcel Dekker New York · Zbl 0684.62071
[6] Hesselager, O.; Wang, S.; Willmot, G., Exponential and scale mixtures and equilibrium distributions, Scandinavian Actuarial Journal (1997), to appear
[7] Hogg, R.; Klugman, S., Loss Distributions (1984), Wiley: Wiley New York
[8] Kaas, R.; van Heerwaarden, A.; Goovaerts, M., Ordering of Actuarial Risks (1994), Ceuterick: Ceuterick Leuven
[9] Launer, R., Inequalities for NBUE and NWUE life distributions, Operations Research, 32, 660-667 (1984) · Zbl 0554.62083
[10] Lin, X., Tail of compound distributions and excess time, Journal of Applied Probability, 33, 184-195 (1996) · Zbl 0848.60081
[11] Massey, W.; Whitt, W., A probabilistic generalization of Taylor’s theorem, Statistics and Probability Letters, 16, 51-54 (1993) · Zbl 0765.60032
[12] Nanda, A.; Jain, K.; Singh, H., Properties of moments for \(s\)-order equilibrium distributions, Journal of Applied Probability, 33, 1108-1111 (1996) · Zbl 0871.60062
[13] Panjer, H., Recursive evaluation of a family of compound distributions, Astin Bulletin, 12, 22-26 (1981)
[14] Pellerey, F., On the preservation of some orderings of risks under convolution, Insurance: Mathematics and Economics, 16, 81-83 (1996) · Zbl 0894.62113
[15] Singh, H., On partial ordering of life distribution, Naval Research Logistics, 36, 103-110 (1989) · Zbl 0658.62116
[16] Tijms, H., Stochastic Modelling and Analysis: A Computational Approach (1986), Wiley: Wiley Chichester, UK
[17] Willmot, G., Refinements and distributional generalizations of Lundberg’s inequality, Insurance: Mathematics and Economics, 15, 49-63 (1994) · Zbl 0814.62070
[18] Willmot, G.; Lin, X., Lundberg bounds on the tails of compound distributions, Journal of Applied Probability, 31, 743-756 (1994) · Zbl 0812.60084
[19] Willmot, G.; Lin, X., Simplified bounds on the tails of compound distributions, Journal of Applied Probability, 34, 127-133 (1997) · Zbl 0882.60088
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