zbMATH — the first resource for mathematics

Best-possible bounds on sets of bivariate distribution functions. (English) Zbl 1057.62038
The aim of this paper is to present a method for finding bounds on arbitrary sets of joint distribution functions $$H$$ of continuous random variables $$X$$ and $$Y$$, $$H$$ having the marginal distribution functions $$F$$ and $$G$$, respectively. The paper uses copulas $$C$$ (uniquely determined by the relation $$H(x, y)= C(F(x),G(y)))$$ and quasi-copulas to narrow the classical Fréchet-Hoeffding inequality (and bounds) in order to obtain pointwise best possible bounds on non-empty sets of distribution functions. As an application, the proposed procedure is illustrated by finding bounds when the values of $$H$$ are known at quartiles of $$X$$ and $$Y$$.

MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 60E15 Inequalities; stochastic orderings 62E10 Characterization and structure theory of statistical distributions 62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text:
References:
 [1] Alsina, C; Nelsen, R.B; Schweizer, B, On the characterization of a class of binary operations on distribution functions, Statist. probab. lett., 17, 85-89, (1993) · Zbl 0798.60023 [2] Fréchet, M, Sur LES tableaux de corrélation dont LES marges sont données, Ann. univ. Lyon sect. A, 9, 53-77, (1951) · Zbl 0045.22905 [3] Fredricks, G.A; Nelsen, R.B, Copulas constructed from diagonal sections, (), 129-136 · Zbl 0906.60022 [4] Fredricks, G.A; Nelsen, R.B, Diagonal copulas, (), 121-128 · Zbl 0906.60021 [5] Fredricks, G.A; Nelsen, R.B, The bertino family of copulas, (), 81-92 · Zbl 1135.62334 [6] Genest, C; Quesada Molina, J.J; Rodrı́guez Lallena, J.A; Sempi, C, A characterization of quasi-copulas, J. multivariate anal., 69, 193-205, (1999), doi:10.1006/ jmva.1998.1809 · Zbl 0935.62059 [7] W. Hoeffding, Masstabinvariante Korrelationstheorie, Schriften des Matematischen Instituts und des Instituts für Angewandte Mathematik der Universität Berlin 5, Heft 3 (1940), 179-233 [Reprinted as Scale-invariant correlation theory in: N.I. Fisher, P.K. Sen (Eds.), The Collected Works of Wassily Hoeffding, Springer, New York, 1994, pp. 57-107]. [8] Nelsen, R.B, An introduction to copulas, (1999), Springer New York · Zbl 0909.62052 [9] Nelsen, R.B; Quesada Molina, J.J; Schweizer, B; Sempi, C, Derivability of some operations on distribution functions, (), 233-243 [10] Nelsen, R.B; Quesada Molina, J.J; Rodrı́guez Lallena, J.A; Úbeda Flores, M, Bounds on bivariate distribution functions with given margins and measures of association, Comm. statist.-theory methods, 30, 1155-1162, (2001) · Zbl 1008.62605 [11] Sklar, A, Fonctions de répartition à n dimensions et leurs marges, Publ. inst. statist. univ. Paris, 8, 229-231, (1959) · Zbl 0100.14202 [12] M. Úbeda Flores, Cópulas y cuasicópulas: interrelaciones y nuevas propiedades, Aplicaciones, Ph.D. Dissertation, Servicio de Publicaciones de la Universidad de Almerı́a, Spain, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.