Best-possible bounds on sets of bivariate distribution functions

By Roger B. Nelsen, J. Q. Molina, J. A. R. Lallena, Manuel Úbeda Flores. 2004

103 citations recorded by the source corpus.

The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Frechet-Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G, respectively, then max(0, F(x) + G(y) - 1)≤H(x,y)≤min(F(x), G(y)) for all x,y in [-∞, ∞]. In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function H with given margins F and G when the values of H are known at quartiles of X and Y.

Subfields: Management Science and Operations Research, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/844f5e68554b6bd524e16488fefc35772700c4d6

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In Management Science and Operations Research

In Numerical Analysis

Citations in the corpus

PARTIAL IDENTIFICATION OF THE DISTRIBUTION OF TREATMENT EFFECTS AND ITS CONFIDENCE SETS (2009, methodology): “In the literature on program evaluation with selection-on-observables, the following two assumptions are often used to evaluate the effect of a treatment or a program, see for example, Rosenbaum and Rubin (1983), Hahn (1998), Heckman, Ichimura, Smith, and Todd (1998), Dehejia and Wahba (1999), and H”
Copula Theory: An Introduction (2010, background): “’s (see [15, 153, 168]), especially when some additional information is given, like lower dimensional marginals [37, 45, 103], and measures of association [151, 155].”
Aggregation Functions: A Guide for Practitioners (2007, background): “Interpolatory constructions are very recent, they have been studied in [15, 17, 19, 25, 26, 27, 67, 109, 140, 150, 197].”
Improved Fréchet Bounds and Model-Free Pricing of Multi-Asset Options (2010, unknown): “This generalizes the findings of [11] on the improved Fréchet bounds for copulas with given diagonal sections.”
A new class of symmetric bivariate copulas (2006, methodology): “[11], and has recently been used to find the best possible bounds on arbitrary sets of bivariate distribution functions ([12]).”
On the construction of copulas and quasi-copulas with given diagonal sections (2008, methodology): “It is known (Bertino, 1977; Fredricks and Nelsen, 2002; Nelsen and Fredricks, 1997; Nelsen et al., 2004) that Cδ and Qδ are nonempty sets for any diagonal δ.”
Copulas: A Review and Recent Developments (2006, methodology): “Bivariate Probability Integral Transform and Related Kendall Distribution Functions An extension of probability integral transform is given by Nelsen et al.[92].”
Copulas in Econometrics (2014, background): “Nelsen & Ubeda-Flores (2004) and Nelsen et al. (2001, 2004) establish improved Fréchet-Hoeffding bounds when such partial dependence information is available.”
Actuarial Theory for Dependent Risks: Measures, Orders and Models (2005, background): “Bivariate positive quadrant dependence has been generalized to higher dimensions in several ways; see Newman (1984). The random vector X is said to be pairwise PQD if Xi and Xj are PQD for all i = j.”
Worst VaR scenarios with given marginals and measures of association (2009, methodology): “Other work related to the problem of Kolmogorov includes, without being exhaustive, Denuit et al. (1999), Nelsen et al. (2001, 2004), Dhaene et al. (2002), Nelsen andÚbeda Flores (2004), Denuit et al. (2005), Embrechts et al. (2005), Embrechts and Puccetti (2006) and Laeven (2009).”