The geometry of certain fixed marginal probability distributions (Q1076460)

\textit{E. L. Lehman} [Ann. Math. Stat. 37, 1137-1153 (1966; Zbl 0146.406)] introduced the concept of two random variables X and Y being positively quadrant dependent (PQD) by \[ P\{X\leq x,\quad Y\leq y\}\geq P\{X\leq x\}P\{Y\leq y\} \] for all (x,y) and of being negatively quadrant dependent (NQD) by \[ P\{X\leq x,\quad Y\leq y\}\leq P\{X\leq x\}P\{Y\leq y\}. \] An extension to orderings can be given in terms of the cumulative distribution function (cdf). Thus, cdf \(F_ 1\) is more concordant than a cdf \(F_ 2\) if \(F_ 1(x,y)\geq F_ 2(x,y)\) for all (x,y). In this case we write \(F_ 1\to^{c}F_ 2\). As a consequence, we can say that the random variables corresponding to \(F_ 1\) are more highly positively dependent than the random variables corresponding to \(F_ 2\). Ordering of the measures of association such as Pearson's and Kendall's rank correlations follows. The present paper is concerned with a geometrical representation of the condition of concordance. If the marginals are fixed this is essentially the geometry of \(p\times q\) matrices when corresponding cdf's are more concordant than a given cdf. It should be noted that in the bivariate situation, if \(F_ 1(x,y)\) is the cdf defined as the product of the marginals of X and Y and F(x,y) is the corresponding joint pdf then \[ F(x,y)\to^{c}F_ 1(x,y)\quad if\quad and\quad only\quad if\quad (X,Y)\quad are\quad PQD,\quad and \] \[ F(x,y)\leftarrow^{c}F_ 1(x,y)\quad if\quad and\quad only\quad if\quad (X,Y)\quad are\quad NQD. \] While general study is made of the \(p\times q\) nonnegative matrix from a geometric point of view, special attention is given to the case of \(p=2\), \(q=2,3\). Diagrammatic representation of the latter case is also given.