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Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  04 September 2020

Lu Li ,
Qinyu Wu  and
Tiantian Mao
Lu Li*
Affiliation:
University of Science and Technology of China
Qinyu Wu*
Affiliation:
University of Science and Technology of China
Tiantian Mao*
Affiliation:
University of Science and Technology of China
*
*Postal address: Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, China. Email address: [email protected]
**Email address: [email protected]
***Email address: [email protected]
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Abstract

We investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

Keywords

Reversed hazard rate orderlikelihood ratio orderusual stochastic orderexponentiated generalized gamma distributionexponentiated Pareto distributionoptimal reinsurance

MSC classification

Primary: 60E05: Distributions
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 832 - 852
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Copyright
© Applied Probability Trust 2020

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References

Al-Hussaini, E. K. and Ahsanullah, M. (2015). Exponentiated Distributions (Atlantis Studies in Probability and Statistics 21). Atlantis Press, Paris.Google Scholar
Balakrishnan, N. and Rao, C. R. (1998). Order Statistics: Theory and Methods (Handbook of Statistics 16). Elsevier, New York.Google Scholar
Balakrishnan, N. and Rao, C. R. (1998). Order Statistics: Applications (Handbook of Statistics 17). Elsevier, New York.Google Scholar
Balakrishnan, N. and Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: a review with an emphasis on some recent developments. Prob. Eng. Inf. Sci. 27 (4), 403–443.10.1017/S0269964813000156CrossRefGoogle Scholar
Bernard, C., He, X., Yan, J. A. and Zhou, X. Y. (2015). Optimal insurance design under rank-dependent expected utility. Math. Finance 25, 154–186.CrossRefGoogle Scholar
Cai, J. and Wei, W. (2012). Optimal reinsurance with positively dependent risks. Insurance Math. Econom. 50 (1), 57–63.CrossRefGoogle Scholar
Cai, J., Tan, K. S., Weng, C. and Zhang, Y. (2008). Optimal reinsurance under VaR and CTE risk measures. Insurance Math. Econom. 43 (1), 185–196.10.1016/j.insmatheco.2008.05.011CrossRefGoogle Scholar
Bon, J. L. and PĂltĂnea, E. (2006). Comparison of order statistics in a random sequence to the same statistics with i.i.d. variables. ESAIM Prob. Statist. 10, 1–10.CrossRefGoogle Scholar
Borch, K. (1962). Equilibrium in a reinsurance market. Econometrica 30, 424–444.CrossRefGoogle Scholar
Chateauneuf, A., Cohen, M. and Meilijson, I. (2004). Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model. J. Math. Econom. 40, 547–571.10.1016/S0304-4068(03)00044-2CrossRefGoogle Scholar
Chateauneuf, A., Cohen, M. and Meilijson, I. (2005). More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model. Economic Theory 25 (3), 649–667.CrossRefGoogle Scholar
Cheung, K. C., Sung, K. C. J., Yam, S. C. P. and Yung, S. P. (2014). Optimal reinsurance under general law-invariant risk measures. Scand. Actuarial J. 2014, 72–91.10.1080/03461238.2011.636880CrossRefGoogle Scholar
Chi, Y. and Tan, K. S. (2011). Optimal reinsurance under VaR and CVaR risk measures: a simplified approach. ASTIN Bull. 41, 487–509.Google Scholar
David, H. A. and Nagaraja, H. N. (1970). Order Statistics. John Wiley.Google Scholar
Di Crescenzo, A. (2000). Some results on the proportional reversed hazards model. Statist. Prob. Lett. 50 (4), 313–321.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Finance and Insurance. Springer, Berlin.CrossRefGoogle Scholar
Gupta, R. C. and Gupta, R. D. (2007). Proportional reversed hazard rate model and its applications. J. Statist. Planning Infer. 137 (11), 3525–3536.CrossRefGoogle Scholar
Haidari, A. and Najafabadi, A. (2019). Characterization ordering results for largest order statistics from heterogeneous and homogeneous exponentiated generalized gamma. Prob. Eng. Inf. Sci. 33, 460–470.CrossRefGoogle Scholar
Kalbfleisch, J. D. and Lawless, J. F. (1989). Inference based on retrospective ascertainment: an analysis of the data on transfusion-related AIDS. J. Amer. Statist. Assoc. 84 (406), 360–372.CrossRefGoogle Scholar
Khaledi, B. E., Farsinezhad, S. and Kochar, S. C. (2011). Stochastic comparisons of order statistics in the scale model. J. Statist. Planning Infer. 141 (1), 276–286.CrossRefGoogle Scholar
Kochar, S. (2012). Stochastic comparisons of order statistics and spacings: a review. ISRN Prob. Statist. 2012, 839473, 1–47.CrossRefGoogle Scholar
Kotz, S., Balakrishnan, N. and Johnson, N. L. (2004). Continuous Multivariate Distributions, Vol. 1: Models and Applications. John Wiley.Google Scholar
Mao, T. and Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Prob. Eng. Inf. Sci. 24 (2), 245–262.CrossRefGoogle Scholar
Mao, T., Hu, T. and Zhao, P. (2010). Ordering convolutions of heterogeneous exponential and geometric distributions revisited. Prob. Eng. Inf. Sci. 24 (3), 329–348.10.1017/S026996481000001XCrossRefGoogle Scholar
Marshall, A. W., Olkin, I. and Arnold, B. C. (1979). Inequalities: Theory of Majorization and its Applications (Springer Series in Statistics 143). Academic Press, New York.Google Scholar
Misra, N. and Misra, A. K. (2013). On comparison of reversed hazard rates of two parallel systems comprising of independent gamma components. Statist. Prob. Lett. 83 (6), 1567–1570.Google Scholar
Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliab. 42 (2), 299–302.10.1109/24.229504CrossRefGoogle Scholar
Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37 (4), 436–445.CrossRefGoogle Scholar
Nadarajah, S. (2005). Exponentiated Pareto distributions. Statistics 39 (3), 255–260.CrossRefGoogle Scholar
Nadarajah, S., Jiang, X. and Chu, J. (2017). Comparisons of smallest order statistics from Pareto distributions with different scale and shape parameters. Ann. Operat. Res. 254 (1–2), 191–209.CrossRefGoogle Scholar
Navarro, J. (2016). Stochastic comparisons of generalized mixtures and coherent systems. Test 25 (1), 150–169.10.1007/s11749-015-0443-5CrossRefGoogle Scholar
Pledger, G. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics, pp. 89–113. Academic Press.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders (Springer Series in Statistics). Springer.CrossRefGoogle Scholar
Wang, B. X., Yu, K. and Coolen, F. P. (2015). Interval estimation for proportional reversed hazard family based on lower record values. Statist. Prob. Lett. 98, 115–122.CrossRefGoogle Scholar
Zhao, P. and Balakrishnan, N. (2012). Stochastic comparisons of largest order statistics from multiple-outlier exponential models. Prob. Eng. Inf. Sci. 26 (2), 159–182.CrossRefGoogle Scholar
Zhao, P. and Balakrishnan, N. (2014). A stochastic inequality for the largest order statistics from heterogeneous gamma variables. J. Multivar. Anal. 129, 145–150.10.1016/j.jmva.2014.04.003CrossRefGoogle Scholar
Zhao, P., Zhang, Y. and Qiao, J. (2016). On extreme order statistics from heterogeneous Weibull variables. Statistics 50 (6), 1376–1386.10.1080/02331888.2016.1230859CrossRefGoogle Scholar