Abstract
In order to take into account any possible dependence between alternatives in optimization problems, bivariate characterizations of some well-know univariate stochastic orders have been defined and studied by Shanthikumar and Yao (Adv Appl Probab 23:642–659, 1991). These characterizations gave rise to new stochastic comparisons, commonly called joint stochastic orders, which are equivalent to the original ones under assumption of independence, but are different whenever the variables to be compared are dependent. In this note we provide sufficient conditions on the survival copula describing the dependence among the compared variables such that the standard stochastic orders imply the corresponding joint stochastic orders, and viceversa. Also, simple conditions for joint stochastic orders between the components of random vectors defined through multivariate frailty models are provided.
Similar content being viewed by others
References
Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing. Probability models. To begin with, Silver Spring
Belzunce F, Martinez-Puertas H, Ruiz JM (2011) On optimal allocation of redundant components for series and parallel systems of two dependent components. J Stat Plan Inference 141:3094–3104
Belzunce F, Ortega EM, Pellerey F, Ruiz JM (2007) On ranking and top choices in random utility models with dependent utilities. Metrika 66:197–212
Boland PJ, Proschan F, Tong YL (1988) Moment and geometric probability inequalities arising from arrangement increasing functions. Ann Probab 16:407–413
Durante F (2006) A new class of symmetric bivariate copulas. J Nonparametr Stat 18:499–510
Hollander M, Proschan F, Sethuraman J (1977) Functions decreasing in transportation and their application in ranking problems. Ann Stat 5:722–733
Hua L, Cheung KC (2008) Stochastic orders of scalar products with applications. Insur Math Econ 42: 865–872
Li X, You Y (2012) A note on allocation of portfolio shares of random assets with Archimedean copula. Ann Oper Res (in press). doi:10.1007/s10479-012-1137-y
Ling X, Zhao P, Li X (2012) On a system of nonhomogeneous components sharing a common frailty. J Stat Plan Inference 142:1330–1338
Ma C (2000) Convex ordering for linear combinations of random variables. J Stat Plan Inference 84:11–25
Mulero J, Pellerey F, Rodríguez-Griñolo R (2010) Negative aging and stochastic comparisons of residual lifetimes in multivariate frailty models. J Stat Plan Inference 140:1594–1600
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, Berlin
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York
Shanthikumar JG, Yamazaki G, Sakasegawa H (1991) Characterization of optimal order of servers in a tandem queue with blocking. Ope Res Lett 10:17–22
Shanthikumar JG, Yao DD (1991) Bivariate characterization of some stochastic order relations. Adv Appl Probab 23:642–659
Wienke A (2011) Frailty models in survival analysis. Chapman & Hall/CRC Biostatistics Series, vol 37. CRC Press, Boca Raton
Acknowledgments
We thank the referee for the careful reading of the paper and the useful suggestions on the presentation of the results and their proofs.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pellerey, F., Zalzadeh, S. A note on relationships between some univariate stochastic orders and the corresponding joint stochastic orders. Metrika 78, 399–414 (2015). https://doi.org/10.1007/s00184-014-0509-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-014-0509-5