Download Advances in Probability Distributions with Given Marginals: by G. Dall’aglio (auth.), G. Dall’Aglio, S. Kotz, G. Salinetti PDF

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

As the reader might most likely already finish from theenthusiastic phrases within the first traces of this assessment, this publication can bestrongly advised to probabilists and statisticians who deal withdistributions with given marginals.
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SCHWEIZER I came across a review of a paper by G. D. Makarov entitled "Estimates for the distribution function of the sum of two random variables with given marginal distributions" [42; MR 83c: 60029]. I looked up the paper and my interest in it increased when I learned that its aim was to answer a question that had been posed by Kolmogorov. But I found Makarov's argument somewhat impenetrable and therefore set out to practice what I have just preached. (z) = Kolmogorov's problem is the following: df(X) = F X and Y with F F such that for all and and df(Y) = G, z in R, infP(X + Y < z) and F(z) = supp(X + Y < z), where the infimum and supremum are taken over all possible joint distribution functions H having margins F and G.

V. 's is the copula of Since the binary operations TT X and X and Y, playa promi- nent role in the theory of probabilistic metric spaces, and since TMin = a Min , it is natural to ask: What binary operations on random variables correspond to these operations? The answer is: none. To make this more precise, we need the following: Generally 28 B. 1. f. 's defined on a common probability space, such that df(X) = F, =G df(Y) and df(V(X,Y)) = ~(F,G). 1. Min. Then 'T Let T be any (left-continuous) t-norm other than is not derivable from any function on random variables.

F) If f and g are strictly monotone respectively. f. of X and relation coefficient r. function of Irl. 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).

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