Stochastic dominance and statistical preference for random variables coupled by arbitrary copulas. (English) Zbl 07834350
Summary: Recently, results have been published showing that first order stochastic dominance implies statistical preference and diff-stochastic dominance, when the copula relating the compared variables is either Archimedean, the product copula, or one of the Fréchet-Hoeffding bounds. In the present paper, we rely on known results on multivariate stochastic orders to extend these results and simplify the proofs. The results are expanded in two directions: First, we show that it suffices for the copula to be symmetric. Second, we reveal that first stochastic dominance entails a wider range of stochastic preferences beyond statistical preference and diff-stochastic dominance. We further analyze whether first stochastic dominance implies statistical preference for the case of asymmetric copulas. We observe that, when at least one of the marginal cumulative distribution functions has no discontinuity jumps, the family of asymmetric copulas for which the implication holds is at least as large as the one for which it does not.
MSC:
| 60E15 | Inequalities; stochastic orderings |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
Keywords:
decision under uncertainty; (multivariate) stochastic ordering; first stochastic dominance; statistical preference; copulaReferences:
| [1] | Marquis de Condorcet, N.d. C., Essai sur l’application de l’analyse à la probabilité des decisions rendues à la pluralité des voix |
| [2] | De Schuymer, B.; De Meyer, H.; De Baets, B.; Jenei, S., On the cycle-transitivity of the dice model. Theory Decis., 261-285 (2003) · Zbl 1075.60011 |
| [3] | Couso, I.; Moral, S.; Sánchez, L., The behavioral meaning of the median. Inf. Sci., 127-138 (2015) · Zbl 1414.91103 |
| [4] | Arcones, M. A.; Kvam, P. H.; Samaniego, F. J., Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J. Am. Stat. Assoc., 457, 170-182 (2002) · Zbl 1073.62520 |
| [5] | Boland, P. J.; Singh, H.; Cukic, B., The stochastic precedence ordering with applications in sampling and testing. J. Appl. Probab., 1, 73-82 (2004) · Zbl 1048.60015 |
| [6] | Navarro, J.; Rubio, R., Comparisons of coherent systems using stochastic precedence. Test, 3, 469-486 (2010) · Zbl 1203.60150 |
| [7] | De Santis, E.; Fantozzi, F.; Spizzichino, F., Relations between stochastic orderings and generalized stochastic precedence. Probab. Eng. Inf. Sci., 3, 329-343 (2015) · Zbl 1370.60029 |
| [8] | Montes, I.; Montes, S., Stochastic dominance and statistical preference for random variables coupled by an Archimedean copula or by the Fréchet-Hoeffding upper bound. J. Multivar. Anal., 275-298 (2016) · Zbl 1384.60061 |
| [9] | De Schuymer, B.; De Meyer, H.; De Baets, B., Cycle-transitive comparison of independent random variables. J. Multivar. Anal., 352-373 (2005) · Zbl 1087.60018 |
| [10] | De Meyer, H.; De Schuymer, B.; De Baets, B., On the transitivity of the comonotonic and countermonotonic comparison of random variables. J. Multivar. Anal., 177-193 (2007) · Zbl 1114.60018 |
| [11] | Montes, I.; Salamanca, J. J.; Montes, S., A modified version of stochastic dominance involving dependence. Stat. Probab. Lett. (2020) · Zbl 1450.60014 |
| [12] | Belzunce, F.; Martínez-Riquelme, C., A new stochastic dominance criterion for dependent random variables with applications. Insur. Math. Econ., 165-176 (2023) · Zbl 1507.91197 |
| [13] | Shanthikumar, J. G.; Yao, D. D., Bivariate characterization of some stochastic order relations. Adv. Appl. Probab., 3, 642-659 (1991) · Zbl 0745.62054 |
| [14] | Kreweras, G., Sur une possibilité de rationaliser les intransitivités, 27-32 (1961), Editions du CNRS, Colloques Internationaux du CNRS, “La Decision”: Editions du CNRS, Colloques Internationaux du CNRS, “La Decision” Paris |
| [15] | Fishburn, P. C., Nontransitive measurable utility. J. Math. Psychol., 1, 31-67 (1982) · Zbl 0499.90006 |
| [16] | Sánchez, L.; Costa, N.; Couso, I., Simplified models of remaining useful life based on stochastic orderings. Reliab. Eng. Syst. Saf. (2023) |
| [17] | Durante, F.; Sempi, C., Principles of Copula Theory (2015), CRC Press |
| [18] | Durante, F.; Gianfreda, A.; Ravazzolo, F.; Rossini, L., A multivariate dependence analysis for electricity prices, demand and renewable energy sources. Inf. Sci., 74-89 (2022) |
| [19] | Puccetti, G., Measuring linear correlation between random vectors. Inf. Sci., 1328-1347 (2022) · Zbl 07816967 |
| [20] | Sklar, M., Fonctions de repartition an dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris, 229-231 (1959) · Zbl 0100.14202 |
| [21] | Nelsen, R. B., Extremes of nonexchangeability. Stat. Pap., 329-336 (2007) · Zbl 1110.62071 |
| [22] | Hadar, J.; Russell, W. R., Rules for ordering uncertain prospects. Am. Econ. Rev., 1, 25-34 (1969) |
| [23] | Savage, L. J., The Foundations of Statistics (1972), Courier Corporation · Zbl 0276.62006 |
| [24] | Kreweras, G., Aggregation of preference orderings, 73-79 |
| [25] | Fishburn, P. C., Nontransitive preferences in decision theory. J. Risk Uncertain., 113-134 (1991) · Zbl 0723.90004 |
| [26] | Loomes, G.; Sugden, R., Regret theory: an alternative theory of rational choice under uncertainty. Econ. J., 368, 805-824 (1982) |
| [27] | Ross, S., Stochastic Processes (1983), Wiley: Wiley New York · Zbl 0555.60002 |
| [28] | Stoyan, D.; Daley, D. J., Comparison Methods for Queues and Other Stochastic Models. (1983), Wiley: Wiley New York · Zbl 0536.60085 |
| [29] | Karlin, S.; Novikov, A., Generalized convex inequalities. Pac. J. Math., 1251-1279 (1963) · Zbl 0126.28102 |
| [30] | Sordo, M. A.; de Souza, M. C.; Suárez-Llorens, A., A new variability order based on tail-heaviness. Statistics, 1042-1061 (2015) · Zbl 1382.60042 |
| [31] | López-Díaz, M., A stochastic ordering for random variables with applications. Aust. N. Z. J. Stat., 1, 1-16 (2010) · Zbl 1337.60016 |
| [32] | Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer · Zbl 1111.62016 |
| [33] | del Barrio, E.; Cuesta-Albertos, J. A.; Matrán, C., Some indices to measure departures from stochastic order (2018), arXiv preprint |
| [34] | Nelsen, R. B., An Introduction to Copulas (2006), Springer · Zbl 1152.62030 |
| [35] | Müller, A.; Scarsini, M., Stochastic comparison of random vectors with a common copula. Math. Oper. Res., 4, 723-740 (2001) · Zbl 1082.60504 |
| [36] | Embrechts, P.; Hofert, M., A note on generalized inverses. Math. Methods Oper. Res., 423-432 (2013) · Zbl 1281.60014 |
| [37] | Siburg, K.; Stoimenov, P., Symmetry of functions and exchangeability of random variables. Stat. Pap., 1-15 (2011) · Zbl 1247.60021 |
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