In 1984 Scarsini introduced a set of axioms for measures of concordance of ordered pairs of continuous random variables. We exhibit an extension of these axioms to ordered n-tuples of continuous random variables, n ≥ 2. We derive simple properties of such measures, give examples, and discuss the relation of the extended axioms to multivariate measures of concordance previously discussed in the literature.

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Multivariate measures of concordance

Concordance and consensus

The present paper introduces a group of transformations on the collection of all bivariate copulas. This group contains an involution which is particularly useful since it provides (1) a criterion under which a given symmetric copula can be transformed into an asymmetric one and (2) a condition under which for a given copula the value of every measure of concordance is equal to zero. The group also contains a subgroup which is of particular interest since its four elements preserve symmetry, the order between two copulas and the value of every measure of concordance.

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Bivariate copulas: Transformations, asymmetry and measures of concordance

We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). We apply these functions to the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality by one of our functions when applied to such a set rules out the existence of the joint distribution. We focus on an especially versatile and widely applicable class of pseudo-quasi-metrics called order-distances. We show, in particular, that the Bell-CHSH-Fine inequalties of quantum physics follow from the triangle inequalities for appropriately defined order-distances. We show how certain metric-like functions on jointly distributed random variables (pseudo-quasimetrics introduced in Section 1) can be used in dealing with the problem of selective probabilistic causality (introduced in Section 2), illustrating this on examples taken from behavioral sciences and quantum physics (Section 3). Although most of Section 2 applies to arbitrary pseudo-quasimetrics on jointly distributed random variables, we single out one, termed order-distance, which is especially useful due to its versatility. We discuss examples of other pseudo-quasi-metrics and rules for their construction in Section 4. 1. Order p.q.-metrics Random variables in this paper are understood in the broadest sense, as measurable functions X : Vs → V , no restrictions being imposed on the sample spaces (Vs,Σs, μs) and the induced probability spaces, (V,Σ, μ), with the usual meaning of the terms (sets of values Vs, V , sigmaalgebras Σs,Σ, and probability measures μs, μ). In particular, any set X of jointly distributed random variables (functions on the same sample space) is a random variable, and its induced probability space (or, simply, distribution) X = (V,Σ, μ) is referred to as the joint distribution of its elements. Given a class of random variables X , not necessarily jointly distributed, let X ∗ be the class of distributions X for all X ∈ X . For any class function f∗ : X ∗ → R (reals), the function f : X → R defined by f (X) = f∗ ( X ) is called observable (as it does not depend on sample spaces, typically unobservable). We will conveniently confuse f and f∗ for observable functions, so that if 2000 Mathematics Subject Classification. Primary 60B99, Secondary 81Q99, 91E45.

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Order-distance and other metric-like functions on jointly distributed random variables

Building upon earlier work in which axioms were formulated for multivariate measures of concordance, we examine properties of such measures. In particular,we examine the relations between the measure of concordance of an n-copula and the measures of concordance of the copula’s marginals.

Multivariate measures of concordance for copulas and their marginals

A continuous random vector ( X , Y ) uniquely determines a
copula C : [ 0 , 1 ] 2 → [ 0 , 1 ] such that when the distribution
functions of X and Y are properly composed into C , the
joint distribution function of ( X , Y ) results. A copula is
said to be D 4 -invariant if its mass distribution is
invariant with respect to the symmetries of the unit square.
A D 4 -invariant copula leads naturally to a family of
measures of concordance having a particular form, and all
copulas generating this family are D 4 -invariant. The
construction examined here includes Spearman’s rho and
Gini’s measure of association as special cases.

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Measures of concordance determined by D4-invariant copulas

If W is a fixed, real-valued random variable, then there are simple and easily satisfied conditions under which the function d W , where d W ( X , Y ) = the probability that W separates the real-valued random variables X and Y , turns out to be a metric. The observation was suggested by work done in [1].

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Separation metrics for real-valued random variables

If T is the parallel map associated with a 1-dimensional tessellation automaton, then we say a configuration f is a weak Garden of Eden for T if f has no pre-image under T other than a shift of itself. Let WG(T)= the set of weak Gardens of Eden for T and G(T)= the set of Gardens of Eden (i.e., the set of configurations not in the range of T). Typically members of WG(T)−G(T) satisfy an equation of the form Tf=Smf where Sm is the shift defined by (Smf)(j)=f(j

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Michael D. Taylor

Papers

Measures of concordance determined by D4-invariant copulas

Separation metrics for real-valued random variables

Weak gardens of eden for 1-dimensional tessellation automata