Characterization of conditional covariance and unified theory in the problem of ordering random variables

Under the assumption that a (p+q)-dimensional row vector (Y, X) is elliptically contoured distributed, the conditional covariance of Y given X=x is characterized in the context of correctly ordering the coordinates Y_k's of Y based on X. This is an answer to a conjecture implicit in Portnoy (1982). Moreover some unified theory is presented for the problem of ordering Y_k's based on X. An essential tool is the decreasing in transposition (D. T.) function theory of Hollander et al. (1977, Ann. Statist., 5(4), 722–733).