By G. Dall’aglio (auth.), G. Dall’Aglio, S. Kotz, G. Salinetti (eds.)

*As the reader may most likely already finish from the**enthusiastic phrases within the first traces of this evaluate, this ebook can be**strongly advised to probabilists and statisticians who deal with**distributions with given marginals.***Mededelingen van het Wiskundig Genootschap**

**Read or Download Advances in Probability Distributions with Given Marginals: Beyond the Copulas PDF**

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**Extra resources for Advances in Probability Distributions with Given Marginals: Beyond the Copulas**

**Example text**

Y) l£1 2 . v. f. 's H sequence n {H } Ran Y. is bivariate normal. Y) is a strictly increasing n . 2 •... , and are pairs of continuous respectively. Y). If and A Y then a. s. Y). Y) n n = a(x. Y) • It follows from property (F) that a is a measure of monotone de- pendence. • a rank statistic. The properties (A) - (H) differ from Renyi's original conditions. Renyi did not require the continuity condition (H). to hold if either X functions g; R. f and = fey) or Y = g(X) He required (E) for some Borel-measurable and (F) to hold for all bijections from R into However.

A mapping from in S [0,1]2 and all into u,v ; 0, [0,1] where satisfying: T is a t-norm, B. 3) where Z(a,b) Z(a,l) = in ~ = whenever b < d, T(a,T(b,c)), [0,1]. T(a,b) ~ =a Z(l,a) a < c, It follows readily that Min(a,b), and Z(a,b) = ° otherwise. 1). 5) W(a,b) Note that Both Z W defined on = = and [0,1]2 Z and (0,1,0) and above by the graph of Min are t-norms, as are the funcby ab, Max(a + b - 1,0). W, Prod and Min are copulas as well as t-norms. 2c) are natural requirements and have ready interpretations.

I worked out some of its properties and showed that, in essence, it satisfied most of Renyi's conditions. F. Wolff and we began to work on them together. This led to a preliminary an- nouncement published in 1976 [63], to Holff's doctoral dissertation [79] a year later, and to a joint paper Which appeared in the Annals of Statistics in 1981 [64]. 1. copula CXY . Let In that paper we established the following: X and Then the quantity tions: (A) a(X,Y) is well-defined. (B) a(X,Y) (C) 0 < a(X,Y) < 1.

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