{"title":"Monotonicity of Dependence Concepts from Independent Random Vector into Dependent Random Vector","authors":"Guangpu Chen","volume":33,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":611,"pagesEnd":621,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/14863","abstract":"
When the failure function is monotone, some monotonic reliability methods are used to gratefully simplify and facilitate the reliability computations. However, these methods often work in a transformed iso-probabilistic space. To this end, a monotonic simulator or transformation is needed in order that the transformed failure function is still monotone. This note proves at first that the output distribution of failure function is invariant under the transformation. And then it presents some conditions under which the transformed function is still monotone in the newly obtained space. These concern the copulas and the dependence concepts. In many engineering applications, the Gaussian copulas are often used to approximate the real word copulas while the available information on the random variables is limited to the set of marginal distributions and the covariances. So this note catches an importance on the conditional monotonicity of the often used transformation from an independent random vector into a dependent random vector with Gaussian copulas.<\/p>\r\n","references":"[1] Eunji Lim and P.W. Glynn. 2006. Simulation-based Response Surface\r\nComputation in the Presence of Monotonicity. Proceedings of the\r\n2006 Winter Simulation Conference.\r\n[2] de Rocquigny E. (2007), Structural Reliability under monotony: A\r\nreview of Properties of form and associated simulation methods and a\r\nNew Class of monotonous reliability methods (MRM), submitted to\r\nStructural Safety.\r\n[3] de Rocquigny E. et al (2007), Optimizing Failure Probability\r\nComputation in a Monotonous Reliability Method, submitted to\r\nReliability Engineering and system Safety.\r\n[4] F. Tonon and al. (2000), Reliability analysis of rock mass response by\r\nmeans of Random Set Theory. Reliability Engineering and System\r\nSafety 70 (2000) 263-282.\r\n[5] Philipp Limbourg et al (2008), Accelerated Uncertainty Propagation\r\nin two-level probabilistic studies under monotony, submitted to\r\nReliability Engineering & System Safety.\r\n[6] Xing Jin and Michael C. Fu. 2001. A Large Deviations Analysis of\r\nQuantile Estimation with Application to Value At Risk.\r\nhttp:\/\/hdl.handle.net\/1903\/2301\r\n[7] O.Detlevsen and H.O. Madsen, (June-September, 2007). Structural\r\nReliability Methods, Internet edition 2.3.7,\r\nhttp:\/\/www.web.mek.dtu.dk\/staff\/od\/books.htm.\r\n[8] RE. Melchers, (1999). Structural Reliability Analysis and Prediction.\r\nEdition, John Wiley & Sons.\r\n[9] Freeman, H. (1963). An Introduction to Statistical Inference.\r\nAddison-Wesley, Reading, MA.\r\n[10] Lehmann E.L. (1966). Some Concepts of Dependence. Ann. Math.\r\nStatist. 37 1137-1153.\r\n[11] Easy J.D.; Proschan F. and Walkup D.W. (1967). Association of\r\nRandom Variables, with applications. Ann. Math. Statist. 38 1466-\r\n1474.\r\n[12] Luger Ruschendorf (1981). Characterization of Dependence Concepts\r\nin Normal Distributions. Ann. Inst. Statist. Math. 33 (1981), Part A,\r\n347-359.\r\n[13] Pitt L.D. (1982). Positively Correlated Normal Variables are\r\nAssociated. Ann. Probability 10 496-499.\r\n[14] Rosenblatt, M. (1952). Remarks on a multivariate transformation.\r\nAnn. Math. Stat. 23(3) : 470-472.\r\n[15] Leslie Hogben (1998). Completions of inverse M-matrix patterns.\r\nLinear Algebra and its Applications 282 (1998) 145-160.\r\n[16] Abraham Berman, Naomi Shaked-Monderer (2003). Completely\r\npositive matrices. World Scientific.\r\n[17] M. Burkschat. 2009. Multivariate Dependence of Spacings of\r\nGeneralized Order Statistics. Journal of Multivariate Analysis 100\r\n(2009) 1093-1106.\r\n[18] Liu, P.-L and Der Kiureghian, A. 1986. Multivariate distribution\r\nmodels with prescribed marginals and covariances. Prob. Engineering\r\nMechanics, 1(2): 105-116.\r\n[19] A. Nataf, (1962). Determination des Distribution dont les Marges sont\r\nDonnees, Comptes Rendus de l-Academie des Sciences, Paris, 225,\r\n42-43.\r\n[20] Markham, T.L. (1971). Factorization of completely positive matrices,\r\nMathematical Proceedings of the Cambridge Philosophical Society,\r\n69(01): 53-58.\r\n[21] Gray, L.J. and Wilson, D.G. (1980). Nonnegative factorization of\r\npositive semi-difinite nonnegative matrices, Linear Algebra and Appl.,\r\n31, 119-127.\r\n[22] M.Kaykobad, (1987). On nonnegative factorization of matrices.\r\nLinear Algebra and Appl., 96, 27-33.\r\n[23] J.H. Drew et al. (1994). Completely positive matrices associated with\r\nM-matrices, Linear and Multilinear Algevre 37, 303-304.\r\n[24] Roger B.Nelsen, (2006). An Introduction to Copulas, Springer Series\r\nin Statistics.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 33, 2009"}