Question

What is the lower bound on a bivariate CDF with uniform binary marginals? What is the upper bound on a bivariate CDF with uniform binary marginals? What do these bounds mean? What PDFs do they correspond to?

Answer 1

The upper bound for the bivariate with uniform binary marginals:-

Any bivariate cdf is bounded by the Fréchet-Hoeffding lower and upper bounds.This bound,which is implicit in logit analysis and the Lorenz curve, and can be used in goodness-of-fit assesment. Any random variable can be expanded in terms of some functions related to this bound. The Bayes approach in comparing two proportions can be presented as the problem of choosing a parametric prior distribution which puts mass on the null hypothesis.

the bivariate upper Frechet bound and the maximal Hoeffding correlation are two related expressions which, directly or implicitly, are quite useful in probability and statistics

The importance of being the upper bound in the bivariate family is consider as-

Let X,Y be two random variables with continuous joint cumulative distribution function (cdf) H(x,y) and marginal cdf’s F(x),G(y).Assuming finite variances,Hoeffding proved that the covariance in terms of the cdf’s is given by

-----------(1)

Then he proved that the correlation coefficient.

satisfies the inequality (ρ−≤ρH≤ρ+), where ρ−,ρ+ are the correlation coefficients for the bivariate cdf’s

respectively.

Frechet has proved the inequality for the function of

-------------(2)

Where H− and H+ are the so-called lower and upper Frechet-Hoeffding bounds. If H reaches these bounds then the following functional relations hold between the random variables

The distributions H−,H+and H=FG (stochastic independence) are examples of cdf’s with marginals F,G.The construction of distributions when the marginals are given is a topic of increasing interest by the proceedings edited by Cuadras, Fortiana and Rodriguez-Lallena which states that

Note that H−and H+are related by

and that the p−dimensional generalization of (2) is

where H(x1,...,xp) is a cdf with univariate marginals F1,...,Fp and

However, if p>2,in general only H+is a cdf, Thus we may focus on the Frechet-Hoeffding upper bound.They present some relevant aspects of H+,which may generate any bivariate cdf and is implicit in some statistical problems.

Distributions of upper bounds:-

Hoeffding’s formula was extended by Cuadras as ("Let us suppose that the ranges of X,Y are the intervals [a,b],[c,d]⊂R,respectively. Thus F(a)=G(c)=0, F(b)=G(d)=1. Let α(x),β(y) be two real functions of bounded variation defined on[a,b],[c,d],respectively.")

If α(a)F(a)=β(c)G(c)=0 and the covariance between α(X),β(Y) exists, it can be obtained from

---------(3)

Suppose that the measure dH(x,y) is absolutely continuous with respect to dF(x) dG(y) and that

Then the following diagonal expansion

-----------(4)

exists, where ρk, ak(X), bk(Y) are the canonical correlations and variables. Let us consider the upper bounds as

and the symmetric kernels

Then using (3) and integrating (4), we can obtain the following expansion

which shows the generating power of the upper bounds.Thus we can consider the nested family as

By taking generalized Orthonormal sets of functions (ak)vand (bk) with respect to F and G.It is worth noting that it can exist a non-countable class of canonical correlations and functions