Abstract
In this paper, a weak version of the joint reversed hazard rate order, useful for stochastic comparison of non-independent random variables, has been defined and discussed. In particular, some relationships between the joint weak reversed hazard rate order and the usual reversed hazard rate order are established when the underlying copulas are symmetric.
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References
R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models (Holt, Rinehart, and Winston, New York, 1981).
B. Bassan and E. Spizzichino, “Relations among Univariate Aging, Bivariate Aging and Dependence for Exchangeable Lifetimes”, J.Multivar.Anal. 93, 313–339 (2005).
F. Belzunce, C. Martínez-Riquelme, and J. Mulero, An Introduction to Stochastic Orders (Academic Press, New York, 2015a).
F. Belzunce, C. Martínez-Riquelme, F. Pellerey, and S. Zalzadeh, “Comparison of Hazard Rates for Dependent Random Variables”, Statistics 50, 1–19 (2015b).
H. W. Block, T. H. Savits, and H. Singh, “The Reversed Hazard Rate Function”, Probability in the Engineering and Informational Sciences 12, 69–90 (1998).
N. K. Chandra and D. Roy, “Some Results on Reverse Hazard Rate”, Probability in the Engineering and Informational Sciences 15, 95–102 (2001).
D.W. Cheng and Y. Zhu, “Optimal Order of Servers in a Tandem Queue with General Blocking”, Queueing System 14, 427–437 (1993).
M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas, Actuarial Theory for Dependent Risks (Wiley, Chichester, England, 2005).
D. Desai, V. Mariappan, and M. Sakhardande, “Nature of Reversed Hazard Rate: An Investigation”, International Journal of Performability Engineering 7, 165–171 (2011).
F. Durante and R. Ghiselli Ricci, “Supermigrative Semi-Copulas and Triangular Norms”, Information Sciences 179, 2689–2694 (2009).
F. Durante and R. Ghiselli Ricci, “Supermigrative Copulas and Positive Dependence”, Adv. in Statist. Anal. 96, 327–342 (2012).
M. S. Finkelstein, “On the Reversed Hazard Rate”, Reliability Engineering & System Safety 78, 71–75 (2002).
S. T. Gross and C. Huber-Carol, “Regression Models for Truncated Survival Data”, Scand. J. Statist. 19, 193–213 (1992).
R. D. Gupta, R. C. Gupta, and P. G. Sankaran, “Some Characterization Results Based on Factorization of the (Reversed) Hazard Rate Function”, Commun. Statist.–Theory Methods 33, 3009–3031 (2004).
J. D. Kalbfleisch and J. F. Lawless, “Regression Models for Right Truncated Data with Applications to AIDS Incubation Times and Reporting Lags”, Statistica Sinica 1, 19–32 (1991).
M. Kijima, “Hazard Rate and Reversed Hazard Rate Monotonicities in Continuous Time Markov Chains”, J. Appl. Probab. 35, 545–556 (1998).
M. Kijima and M. Ohnishi, “Portfolio Selection Problems via the Bivariate Characterization of Stochastic Dominance Relations”, Math. Finance 6, 237–277 (1996).
G. Kimeldorf and A. R. Sampson, “A Framework for Positive Dependence”, Ann. Inst. Statist. Math. 41, 31–45 (1989).
A. Müller and M. Scarsini (Eds.), Stochastic Orders and Decision Under Risk, in Lecture Notes–Monograph Series (IMS, Hayward, California, 1991), Vol.19.
A. Müller and D. Stoyan, Comparison Methods for Stochastic Models and Risks (Wiley, Chichester, England, 2002).
R. B. Nelsen, An Introduction to Copulas (Springer, New York, 2006).
M. H. Poursaeed, “A Note on the Mean Past and the Mean Residual Life of a (n -k + 1)-out-of-n System underMultiMonitoring”, Statist. Papers 51, 409–419 (2010).
M. Razmkhah, H. Morabbi, and J. Ahmadi, “Comparing Two Sampling Schemes Based on Entropy of Record Statistics”, Statist. Papers 53, 95–106 (2012).
M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications (Springer, New York, 2006).
J. G. Shanthikumar and D. D. Yao, “Bivariate Characterization of Some Stochastic Order Relations”, Adv. in Appl. Probab. 23, 642–659 (1991).
J. T. Townsend and M. J. Wenger, “A Theory of Interactive Parallel Processing: New Capacity Measures and Predictions for a Response Time Inequality Series”, Psychol.Rev. 111, 1003–1035 (2004).
E. J. Veres-Ferrer and M. P. Jose, “On the Relationship between the Reversed Hazard Rate and Elasticity”, Statist. Papers 55, 275–284 (2014).
M. Xie, O. Gaudoin, and C. Bracquemond, “Redefining Failure Rate Function for Discrete Distributions”, Internat. J. of Reliability, Quality and Safety Engineering 9, 275–285 (2002).
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Balakrishnan, N., Barmalzan, G. & Kosari, S. On joint weak reversed hazard rate order under symmetric copulas. Math. Meth. Stat. 26, 311–318 (2017). https://doi.org/10.3103/S1066530717040056
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DOI: https://doi.org/10.3103/S1066530717040056
Keywords
- joint reversed hazard rate order
- usual reversed hazard rate order
- symmetric copula
- supermigrativity
- submigrativity