Sklar's theorem in an imprecise setting

Sklar's theorem is an important tool that connects bidimensional distribution functions with their marginals by means of a copula. When there is imprecision about the marginals, we can model the available information by means of p-boxes, that are pairs of ordered distribution functions. Similarly, we can consider a set of copulas instead of a single one. We study the extension of Sklar's theorem under these conditions, and link the obtained results to stochastic ordering with imprecision.

[1]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[2]  Roger B. Nelsen,et al.  Best-possible bounds on sets of bivariate distribution functions , 2004 .

[3]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[4]  Marco Scarsini.  Copulae of capacities on product spaces , 1996 .

[5]  Fabio Gagliardi Cozman,et al.  Kuznetsov independence for interval-valued expectations and sets of probability distributions: Properties and algorithms , 2014, Int. J. Approx. Reason..

[6]  Susana Montes,et al.  Comparison of Random Variables Coupled by Archimedean Copulas , 2010, SMPS.

[7]  Diego A. Alvarez,et al.  A Monte Carlo-based method for the estimation of lower and upper probabilities of events using infinite random sets of indexable type , 2009, Fuzzy Sets Syst..

[8]  Fabrizio Durante,et al.  Semi-copulas, capacities and families of level sets , 2010, Fuzzy Sets Syst..

[9]  Susana Montes,et al.  Decision making with imprecise probabilities and utilities by means of statistical preference and stochastic dominance , 2014, Eur. J. Oper. Res..

[10]  M. Fréchet.  Généralisation du théorème des probabilités totales , 1935 .

[11]  B. De Baets,et al.  Cycle-transitive comparison of independent random variables , 2005 .

[12]  Scott Ferson,et al.  Constructing Probability Boxes and Dempster-Shafer Structures , 2003 .

[13]  D. Bunn.  Stochastic Dominance , 1979 .

[14]  Bernhard Schmelzer.  Joint distributions of random sets and their relation to copulas , 2015, Int. J. Approx. Reason..

[15]  Bernard De Baets,et al.  A Fuzzy Approach to Stochastic Dominance of Random Variables , 2003, IFSA.

[16]  Roger B. Nelsen,et al.  The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas , 2005 .

[17]  B. De Baets,et al.  On the Cycle-Transitivity of the Dice Model , 2003 .

[18]  Gert de Cooman,et al.  Independent natural extension , 2010, Artif. Intell..

[19]  Enrique Miranda,et al.  Bivariate p-boxes , 2016, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[20]  Ronald R. Yager,et al.  Joint cumulative distribution functions for Dempster–Shafer belief structures using copulas , 2013, Fuzzy Optim. Decis. Mak..

[21]  R. Nelsen.  An Introduction to Copulas , 1998 .

[22]  Moshe Shaked,et al.  Stochastic orders and their applications , 1994 .

[23]  P. Walley,et al.  A survey of concepts of independence for imprecise probabilities , 2000 .

[24]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[25]  Susana Montes,et al.  Statistical Preference as a Tool in Consensus Processes , 2011, Consensual Processes.

[26]  Didier Dubois,et al.  Qualitative decision theory with preference relations and comparative uncertainty: An axiomatic approach , 2003, Artif. Intell..

[27]  Sébastien Destercke,et al.  Probability boxes on totally preordered spaces for multivariate modelling , 2011, Int. J. Approx. Reason..

[28]  Susana Montes,et al.  Stochastic dominance with imprecise information , 2014, Comput. Stat. Data Anal..