zbMATH — the first resource for mathematics

The complete mixability and convex minimization problems with monotone marginal densities. (English) Zbl 1229.60019
It is shown in this paper that distributions \(P\) with a monotone density on a bounded interval are \(n\)-mixable under a moderate mean conditions, i.e., there exist \(X_i \sim P\), \(1 \leq i \leq n\), with \(\sum_{i=1}^{n} X_i =c\). In the case of symmetric unimodal distributions this property was shown before in [L. Rüschendorf and L. Uckelmann, in: Distributions with given marginals and statistical modelling. Papers presented at the meeting, Barcelona, Spain, July 17–20, 2000. Dordrecht: Kluwer Academic Publishers. 211–222 (2002; Zbl 1142.62316)]. Based on this property the generalized variance minimization problem to minimize \(\operatorname{E} f (X_i + \dots + X_n)\) over \(X_i \sim P\) for any convex function \(f\) is solved for distributions \(P\) with monotone density.

60E05 Probability distributions: general theory
60E15 Inequalities; stochastic orderings
Full Text: DOI
[1] F. Baiocchi, 1991, Un problema di ottimizzatione nella classe di Fréchet. Unpublished Thesis, Facoltà di Scienze Statistiche, Università La Sapienza, Roma.
[2] Bertino, S., The minimum of the expected value of the product of three random variables in the frèchet class, Journal of the Italian statistical society, 201-211, (1994), 1994-1996 · Zbl 1446.60014
[3] Denneberg, D., Non-additive measure and integral, (1994), Kluwer Academic Publishers Boston · Zbl 0826.28002
[4] Embrechts, P.; Lindskog, F.; McNeil, A., Modelling dependence with copulas and applications to risk management, (), 329-384
[5] Embrechts, P.; McNeil, A.; Straumann, D., Correlation and dependence in risk management: properties and pitfalls, (), 176-223
[6] Embrechts, P.; Höing, A.; Juri, A., Using copulae to bound the value-at-risk for functions of dependent risks, Finance and stoch, 7, 145-167, (2003) · Zbl 1039.91023
[7] Embrechts, P.; Puccetti, G., Bounds for functions of dependent risks, Finance and stoch, 10, 341-352, (2006) · Zbl 1101.60010
[8] Embrechts, P.; Puccetti, G., Risk aggregation, ()
[9] Fishman, G.S., Variance reduction in simulation studies, Journal of statistical computation and simulation, 1, 173-182, (1972) · Zbl 0249.62085
[10] Gaffke, N.; Rüschendorf, L., On a class of extremal problems in statistics, Mathamatische operationsforschung und statistik, series optimization, 12, 123-135, (1981) · Zbl 0467.60004
[11] Hammersley, I.M.; Handscomb, D.C., Monte Carlo methods, (1964), Methuen London · Zbl 0121.35503
[12] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[13] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory, (2001), Kluwer Academic Publishers Boston
[14] M. Knott, C.S. Smith, 1998, On multivariate variance reduction, Unpublished results.
[15] Knott, M.; Smith, C.S., Choosing joint distributions so that the variance of the sum is small, Journal of multivariate analysis, 97, 1757-1765, (2006) · Zbl 1099.60014
[16] Nelsen, R., An introduction to copulas, (2006), Springer New York · Zbl 1152.62030
[17] R. Nelsen, M. Úbeda-Flores, 2010, Directional dependence in multivariate distributions, Unpublished results. · Zbl 1237.62076
[18] Rüschendorf, L., Inequalities for the expectation of \(\Lambda\)-monotone functions, Probability and related fields, 54, 341-349, (1980) · Zbl 0441.60013
[19] Rüschendorf, L., Random variables with maximum sums, Advances in applied probability, 14, 623-632, (1982) · Zbl 0487.60026
[20] Rüschendorf, L.; Uckelmann, L., Variance minimization and random variables with constant sum, (), 211-222 · Zbl 1142.62316
[21] Shaked, M.; Shanthikumar, J.G., ()
[22] Yang, J.; Qi, Y.; Wang, R., A class of multivariate copulas with bivariate Fréchet marginal copulas, Insurance: mathematics and economics, 45, 1, 139-147, (2009) · Zbl 1231.91253
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.