A note on the matrix ordering of special C-matrices (Q1082754)

The matrix inequality (*) \(\Delta_ t-tt'\leq \Delta_ r-rr'\) is considered, where t and r are positive stochastic vectors, and \(\Delta_ t\) and \(\Delta_ r\) are diagonal matrices with t and r on their diagonals. Necessary and sufficient conditions are established (1) for (*) to hold when t and r are given, and (2) for the existence of some vector r satisfying (*) when t is given. The results have applications in various parts of statistics.