Monotonicity of Dependence Concepts from Independent Random Vector into Dependent Random Vector
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.
[1] Eunji Lim and P.W. Glynn. 2006. Simulation-based Response Surface
Computation in the Presence of Monotonicity. Proceedings of the
2006 Winter Simulation Conference.
[2] de Rocquigny E. (2007), Structural Reliability under monotony: A
review of Properties of form and associated simulation methods and a
New Class of monotonous reliability methods (MRM), submitted to
Structural Safety.
[3] de Rocquigny E. et al (2007), Optimizing Failure Probability
Computation in a Monotonous Reliability Method, submitted to
Reliability Engineering and system Safety.
[4] F. Tonon and al. (2000), Reliability analysis of rock mass response by
means of Random Set Theory. Reliability Engineering and System
Safety 70 (2000) 263-282.
[5] Philipp Limbourg et al (2008), Accelerated Uncertainty Propagation
in two-level probabilistic studies under monotony, submitted to
Reliability Engineering & System Safety.
[6] Xing Jin and Michael C. Fu. 2001. A Large Deviations Analysis of
Quantile Estimation with Application to Value At Risk.
http://hdl.handle.net/1903/2301
[7] O.Detlevsen and H.O. Madsen, (June-September, 2007). Structural
Reliability Methods, Internet edition 2.3.7,
http://www.web.mek.dtu.dk/staff/od/books.htm.
[8] RE. Melchers, (1999). Structural Reliability Analysis and Prediction.
Edition, John Wiley & Sons.
[9] Freeman, H. (1963). An Introduction to Statistical Inference.
Addison-Wesley, Reading, MA.
[10] Lehmann E.L. (1966). Some Concepts of Dependence. Ann. Math.
Statist. 37 1137-1153.
[11] Easy J.D.; Proschan F. and Walkup D.W. (1967). Association of
Random Variables, with applications. Ann. Math. Statist. 38 1466-
1474.
[12] Luger Ruschendorf (1981). Characterization of Dependence Concepts
in Normal Distributions. Ann. Inst. Statist. Math. 33 (1981), Part A,
347-359.
[13] Pitt L.D. (1982). Positively Correlated Normal Variables are
Associated. Ann. Probability 10 496-499.
[14] Rosenblatt, M. (1952). Remarks on a multivariate transformation.
Ann. Math. Stat. 23(3) : 470-472.
[15] Leslie Hogben (1998). Completions of inverse M-matrix patterns.
Linear Algebra and its Applications 282 (1998) 145-160.
[16] Abraham Berman, Naomi Shaked-Monderer (2003). Completely
positive matrices. World Scientific.
[17] M. Burkschat. 2009. Multivariate Dependence of Spacings of
Generalized Order Statistics. Journal of Multivariate Analysis 100
(2009) 1093-1106.
[18] Liu, P.-L and Der Kiureghian, A. 1986. Multivariate distribution
models with prescribed marginals and covariances. Prob. Engineering
Mechanics, 1(2): 105-116.
[19] A. Nataf, (1962). Determination des Distribution dont les Marges sont
Donnees, Comptes Rendus de l-Academie des Sciences, Paris, 225,
42-43.
[20] Markham, T.L. (1971). Factorization of completely positive matrices,
Mathematical Proceedings of the Cambridge Philosophical Society,
69(01): 53-58.
[21] Gray, L.J. and Wilson, D.G. (1980). Nonnegative factorization of
positive semi-difinite nonnegative matrices, Linear Algebra and Appl.,
31, 119-127.
[22] M.Kaykobad, (1987). On nonnegative factorization of matrices.
Linear Algebra and Appl., 96, 27-33.
[23] J.H. Drew et al. (1994). Completely positive matrices associated with
M-matrices, Linear and Multilinear Algevre 37, 303-304.
[24] Roger B.Nelsen, (2006). An Introduction to Copulas, Springer Series
in Statistics.
[1] Eunji Lim and P.W. Glynn. 2006. Simulation-based Response Surface
Computation in the Presence of Monotonicity. Proceedings of the
2006 Winter Simulation Conference.
[2] de Rocquigny E. (2007), Structural Reliability under monotony: A
review of Properties of form and associated simulation methods and a
New Class of monotonous reliability methods (MRM), submitted to
Structural Safety.
[3] de Rocquigny E. et al (2007), Optimizing Failure Probability
Computation in a Monotonous Reliability Method, submitted to
Reliability Engineering and system Safety.
[4] F. Tonon and al. (2000), Reliability analysis of rock mass response by
means of Random Set Theory. Reliability Engineering and System
Safety 70 (2000) 263-282.
[5] Philipp Limbourg et al (2008), Accelerated Uncertainty Propagation
in two-level probabilistic studies under monotony, submitted to
Reliability Engineering & System Safety.
[6] Xing Jin and Michael C. Fu. 2001. A Large Deviations Analysis of
Quantile Estimation with Application to Value At Risk.
http://hdl.handle.net/1903/2301
[7] O.Detlevsen and H.O. Madsen, (June-September, 2007). Structural
Reliability Methods, Internet edition 2.3.7,
http://www.web.mek.dtu.dk/staff/od/books.htm.
[8] RE. Melchers, (1999). Structural Reliability Analysis and Prediction.
Edition, John Wiley & Sons.
[9] Freeman, H. (1963). An Introduction to Statistical Inference.
Addison-Wesley, Reading, MA.
[10] Lehmann E.L. (1966). Some Concepts of Dependence. Ann. Math.
Statist. 37 1137-1153.
[11] Easy J.D.; Proschan F. and Walkup D.W. (1967). Association of
Random Variables, with applications. Ann. Math. Statist. 38 1466-
1474.
[12] Luger Ruschendorf (1981). Characterization of Dependence Concepts
in Normal Distributions. Ann. Inst. Statist. Math. 33 (1981), Part A,
347-359.
[13] Pitt L.D. (1982). Positively Correlated Normal Variables are
Associated. Ann. Probability 10 496-499.
[14] Rosenblatt, M. (1952). Remarks on a multivariate transformation.
Ann. Math. Stat. 23(3) : 470-472.
[15] Leslie Hogben (1998). Completions of inverse M-matrix patterns.
Linear Algebra and its Applications 282 (1998) 145-160.
[16] Abraham Berman, Naomi Shaked-Monderer (2003). Completely
positive matrices. World Scientific.
[17] M. Burkschat. 2009. Multivariate Dependence of Spacings of
Generalized Order Statistics. Journal of Multivariate Analysis 100
(2009) 1093-1106.
[18] Liu, P.-L and Der Kiureghian, A. 1986. Multivariate distribution
models with prescribed marginals and covariances. Prob. Engineering
Mechanics, 1(2): 105-116.
[19] A. Nataf, (1962). Determination des Distribution dont les Marges sont
Donnees, Comptes Rendus de l-Academie des Sciences, Paris, 225,
42-43.
[20] Markham, T.L. (1971). Factorization of completely positive matrices,
Mathematical Proceedings of the Cambridge Philosophical Society,
69(01): 53-58.
[21] Gray, L.J. and Wilson, D.G. (1980). Nonnegative factorization of
positive semi-difinite nonnegative matrices, Linear Algebra and Appl.,
31, 119-127.
[22] M.Kaykobad, (1987). On nonnegative factorization of matrices.
Linear Algebra and Appl., 96, 27-33.
[23] J.H. Drew et al. (1994). Completely positive matrices associated with
M-matrices, Linear and Multilinear Algevre 37, 303-304.
[24] Roger B.Nelsen, (2006). An Introduction to Copulas, Springer Series
in Statistics.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:63862", author = "Guangpu Chen", title = "Monotonicity of Dependence Concepts from Independent Random Vector into Dependent Random Vector", 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.
", keywords = "Monotonic, Rosenblatt, Nataf transformation, dependence concepts, completely positive matrices, Gaussiancopulas", volume = "3", number = "9", pages = "742-10", }
