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

As the reader may most likely already finish from theenthusiastic phrases within the first traces of this evaluate, this ebook can bestrongly advised to probabilists and statisticians who deal withdistributions with given marginals.
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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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