Equivalence of Hartley—David—Gumbel and Papathanasiou bounds and some further remarks

Let X1:2, X2:2 be the order statistics of a sample of size two from an absolutely continuous distribution F with finite variance. In this note, we show that the upper bound for Cov(X1:2, X2:2) (and the associated characterization of the uniform distribution) established recently by Papathanasiou (1990) is equivalent to the well-known upper bound for E(X2:2) due to Hartley and David (1954) and Gumbel (1954). This simple argument is also shown to …