Abstract
The Marshall-Olkin family of probability distributions has been the inspiration of numerous research publications in the field of probability distributions. In this paper, we present several new properties of this family. In particular, we focus on stochastic orders, stress-strength reliability, Lorenz and the Leimkhuler curves, compounding, and integrated tail distribution. Two applications related to Lorenz curves and ruin theory are finally presented.
Acknowledgement
The authors would like to thank the anonymous referees for valuable comments and suggestions which improved the presentation of this paper.
EGD was partially funded by grant PID2021-127989OB-I00 (Ministerio de Economía y Competitividad, Spain) and by grant TUR-RETOS2022-075 (Ministerio de Industria, Comercio y Turismo).
Communicated by Gejza Wimmer
References
[1] Ahmad, H. A. H.—Almetwally, E. M.: Marshall-Olkin generalized Pareto distribution: Bayesian and non Bayesian estimation, Pak. J. Stat. Oper. Res. 16 (2020), 21–33.10.18187/pjsor.v16i1.2935Suche in Google Scholar
[2] Alshangiti, A. M.—Kayid, M.—Alarfaj, B.: A new family of Marshall-Olkin extended distributions, J. Comput. Appl. Math. 271(1) (2014), 369–379.10.1016/j.cam.2014.04.020Suche in Google Scholar
[3] Balakrishnan, N.—Sarabia, J.—Kolev, N.: A simple relation between the Leimkuhler curve and the mean residual life, J. Informetr. 4(4) (2010), 602–607.10.1016/j.joi.2010.06.009Suche in Google Scholar
[4] Barakat, H. M.—Ghitany, M. E.—Al-Hussaini, E. K.: Asymptotic distributions of order statistics and record values under the Marshall-Olkin parameterization operator, Comm. Statist. Theory Methods 38 (2009), 2267–2273.10.1080/03610920802361373Suche in Google Scholar
[5] Barreto-Souza, W.—Lemonte, A. J.—Cordeiro, G. M.: General results for the Marshall and Olkin’s family of distributions, An. Acad. Brasil. Ciênc. 85(1) (2013), 3–21.10.1590/S0001-37652013000100002Suche in Google Scholar
[6] Bryson, M.: Heavy Tailed Distributions: Properties and Tests, Technometrics 16(1) (1974), 61–68.10.1080/00401706.1974.10489150Suche in Google Scholar
[7] Burrell, Q. L.: Symmetry and other transformation features of Lorenz/Leimkuhler representations of informetric data, Inf. Process. Manag. 41 (2005), 1317–1329.10.1016/j.ipm.2005.03.016Suche in Google Scholar
[8] Dickson, D.: Insurance Risk and Ruin, Cambridge University Press, Cambridge, 2005.10.1017/CBO9780511624155Suche in Google Scholar
[9] Embrechts, P.—Goldie, C. M.: On closure and factorization properties of subexponential and related distributions, J. Aust. Math. Soc. 29 (1980), 243–256.10.1017/S1446788700021224Suche in Google Scholar
[10] Foss, S.—Korshunov, D.—Zachary, S.: An Introduction to Heavy-Tailed and Subexponential Distributions. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2011.10.1007/978-1-4419-9473-8Suche in Google Scholar
[11] García, V.—Gómez-Déniz, E.—Vázquez-Polo, F. J.: A new skew generalization of the Normal distribution: properties and applications, Comput. Statist. Data Anal. 54 (2010), 2021–2034.10.1016/j.csda.2010.03.003Suche in Google Scholar
[12] Ghitany, M. E.: Marshall-Olkin extended Pareto distribution and its application, Int. J. Appl. Math. Comput. Sci. 18 (2005), 17–31.Suche in Google Scholar
[13] Ghitany, M. E.—Al-Awadhi, F. A.—Alkhalfan, L. A.: Marshall-Olkin extended Lomax distribution and its application to censored data, Comm. Statist. Theory Methods 36 (2007), 1855–1866.10.1080/03610920601126571Suche in Google Scholar
[14] Ghitany, M. E.—Kotz, S.: Reliability properties of extended linear failure-rate distribution, Probab. Engrg. Inform. Sci. 21 (2007), 441–450.10.1017/S0269964807000071Suche in Google Scholar
[15] Jessen, A.—Mikosch, T.: Regularly varying functions, Publ. Inst. Math. 80(94) (2006), 171–192.10.2298/PIM0694171JSuche in Google Scholar
[16] Kleiber, C.—Kotz, S.: Statistical Size Distributions in Economics and Actuarial Sciences, John Wiley & Sons, Inc., 2003.10.1002/0471457175Suche in Google Scholar
[17] Klüppelberg, C.: Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988), 132–141.10.1017/S0021900200040705Suche in Google Scholar
[18] Konstantinides, D.: Risk Theory. A Heavy Tail Approach, World Scientific Publishing, 2018.10.1142/10523Suche in Google Scholar
[19] Kotz, S.—Kumelskii, Y.—Pensky, M.: The Stress-Strenght Model and its Generalizations, World Scientific Publising, 2003.10.1142/5015Suche in Google Scholar
[20] Marshall, A. W.–Olkin, I.: A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika 84(3) (1997), 641–652.10.1093/biomet/84.3.641Suche in Google Scholar
[21] Okasha, H. M.—El-Baz, A. H.—Basheer A. M.: On Marshall-Olkin extended inverse Weibull distribution: properties and estimation using type-II censoring data, J. Stat. Appl. Probab. 7(1) (2020), 9–21.10.18576/jsapl/070102Suche in Google Scholar
[22] Rolski, T.—Schmidli, H.—Schmidt, V.—Teugel, J.: Stochastic Processes for Insurance and Finance, John Wiley & Sons, 1999.10.1002/9780470317044Suche in Google Scholar
[23] Ryu, H. K.—Slottje, D. J.: Two flexible functional form approaches for approximating the Lorenz curve, J. Econometrics 72 (1996), 251–274.10.1016/0304-4076(94)01722-0Suche in Google Scholar
[24] Sarabia, J. M.—Castillo, E.: About a class of max-stable families with applications to income distributions, Metron LXIII 3 (2005), 505–527.Suche in Google Scholar
[25] Sarabia, J. M.—Gómez-Déniz, E.—Sarabia, M.—Prieto, F.: A general method for generating parametric Lorenz and Leimkuhler curves, J. Informetr. 4 (2010), 424–539.10.1016/j.joi.2010.06.002Suche in Google Scholar
[26] Shaked, M.—Shanthikumar, J. G.: Stochastic Orders. Series: Springer Series in Statistics, Springer, 2007.10.1007/978-0-387-34675-5Suche in Google Scholar
© 2024 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets
Artikel in diesem Heft
- 10.1515/ms-2024-frontmatter4
- Intervals of posets of a zero-divisor graph
- Coalgebraic methods for Ramsey degrees of unary algebras
- On nonexistence of D(n)-quadruples
- Rees short exact sequences and preenvelopes
- Generalized discrete Grüss and related results with applications
- Radius problem associated with certain ratios and linear combinations of analytic functions
- Existence results for a fourth order problem with functional perturbed clamped beam boundary conditions
- Oscillatory and asymptotic behavior of even-order nonlinear differential equations with mixed neutral terms
- On a solvable four-dimensional system of difference equations
- Euclidean operator radius inequalities of d-tuple operators and operator matrices
- Equable parallelograms on the Eisenstein lattice
- On certain star versions of the Hurewicz property using ideals
- Relative versions of star-Menger property
- The Maxwell-Boltzmann-Exponential distribution with regression model
- New results for the Marshall-Olkin family of distributions
- A new family of copulas based on probability generating functions
- Induced mappings on the hyperspace of totally disconnected sets