zbMATH — the first resource for mathematics

Characterizations on heavy-tailed distributions by means of hazard rate. (English) Zbl 1043.60012
Summary: Let \(F(x)\) be a distribution function supported on \([0,\infty)\), with an equilibrium distribution function \(F_e(x)\). We study the function \[ r_e(x)= (-\ln \overline{F}_e(x))'= \overline{F}(x) \biggl/ \int_x^\infty \overline{F}(u)\, du, \] which is called the equilibrium hazard rate of \(F\). By the limiting behavior of \(r_e(x)\) we give a criterion to identify \(F\) to be heavy-tailed or light-tailed. Two broad classes of hevy-tailed distributions are also introduced and studied.

60E05 Probability distributions: general theory
62E99 Statistical distribution theory
Full Text: DOI
[1] Asmussen, S. Ruin Probabilities. World Scientific, Singapore, 2000
[2] Embrechts, P., Klüppelberg, C., Mikosch, T. Modelling extremal events for insurance and finance. Springer-Verlag, Berlin, 1997 · Zbl 0873.62116
[3] Embrechts, P., Omey, E. A property of long-tailed distribution. J. Appl. Probab., 21: 80–87 (1984) · Zbl 0534.60015 · doi:10.2307/3213666
[4] Jelenković, P.R., Lazar, A.A. Asymptotic results for multiplexing subexponential on-off processes. Adv. Appl. Prob., 31: 394–421 (1999) · Zbl 0952.60098 · doi:10.1239/aap/1029955141
[5] Klüppelberg, C. Subexponential distributions and integrated tails. J. Appl. Probab., 25: 132–141 (1988) · Zbl 0651.60020 · doi:10.2307/3214240
[6] Pitman, E.J.G. Subexponential distribution functions. J. Austral. Math. Soc. (Series A), 29: 337–347 (1980) · Zbl 0425.60012 · doi:10.1017/S1446788700021340
[7] Rolski, T., Schmidli, H., Schmidt, V., Teugels, J. Stochastic processes for insurance and finance. John Wiley & Sons, Chichester, England, 1999 · Zbl 0940.60005
[8] Schlegel, S. Ruin probabilities in perturbed risk models. Insurance: Mathematics and Economics, 22(1): 93–104 (1998) · Zbl 0907.90100 · doi:10.1016/S0167-6687(98)00011-0
[9] Seneta, E. Functions of regular variation. Lecture Notes in Mathematics 506, Springer-Verlag, New York, 1976 · Zbl 0324.26002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.