On the Comonotone Natural Extension of Marginal p-Boxes

Ignacio Montes
Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, PMLR 147:210-219, 2021.

Abstract

The relationship between several random variables is gathered by their joint distribution. While this distribution can be easily determined by the marginals when an assumption of independence is satisfied, there are situations where the random variables are connected by some dependence structure. One such structure that arises often in practice is comonotonicity. This type of dependence refers to random variables that increase or decrease simultaneously. This paper studies the property of comonotonicity when the uncertainty about the random variables is modelled using p-boxes and the induced coherent lower probabilities. In particular, we analyse the problem of finding a comonotone lower probability with given marginal p-boxes, focusing on the existence, construction and uniqueness of such a model. Also, we prove that, under some conditions, there is a most conservative comonotone lower probability with the given marginal p-boxes, that will be called the comonotone natural extension.

Cite this Paper

BibTeX
@InProceedings{pmlr-v147-montes21a, title = {On the Comonotone Natural Extension of Marginal p-Boxes}, author = {Montes, Ignacio}, booktitle = {Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications}, pages = {210--219}, year = {2021}, editor = {Cano, Andrés and De Bock, Jasper and Miranda, Enrique and Moral, Serafı́n}, volume = {147}, series = {Proceedings of Machine Learning Research}, month = {06--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v147/montes21a/montes21a.pdf}, url = {https://proceedings.mlr.press/v147/montes21a.html}, abstract = {The relationship between several random variables is gathered by their joint distribution. While this distribution can be easily determined by the marginals when an assumption of independence is satisfied, there are situations where the random variables are connected by some dependence structure. One such structure that arises often in practice is comonotonicity. This type of dependence refers to random variables that increase or decrease simultaneously. This paper studies the property of comonotonicity when the uncertainty about the random variables is modelled using p-boxes and the induced coherent lower probabilities. In particular, we analyse the problem of finding a comonotone lower probability with given marginal p-boxes, focusing on the existence, construction and uniqueness of such a model. Also, we prove that, under some conditions, there is a most conservative comonotone lower probability with the given marginal p-boxes, that will be called the comonotone natural extension.} }
Endnote
%0 Conference Paper %T On the Comonotone Natural Extension of Marginal p-Boxes %A Ignacio Montes %B Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2021 %E Andrés Cano %E Jasper De Bock %E Enrique Miranda %E Serafı́n Moral %F pmlr-v147-montes21a %I PMLR %P 210--219 %U https://proceedings.mlr.press/v147/montes21a.html %V 147 %X The relationship between several random variables is gathered by their joint distribution. While this distribution can be easily determined by the marginals when an assumption of independence is satisfied, there are situations where the random variables are connected by some dependence structure. One such structure that arises often in practice is comonotonicity. This type of dependence refers to random variables that increase or decrease simultaneously. This paper studies the property of comonotonicity when the uncertainty about the random variables is modelled using p-boxes and the induced coherent lower probabilities. In particular, we analyse the problem of finding a comonotone lower probability with given marginal p-boxes, focusing on the existence, construction and uniqueness of such a model. Also, we prove that, under some conditions, there is a most conservative comonotone lower probability with the given marginal p-boxes, that will be called the comonotone natural extension.
APA
Montes, I.. (2021). On the Comonotone Natural Extension of Marginal p-Boxes. Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 147:210-219 Available from https://proceedings.mlr.press/v147/montes21a.html.

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