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 yield immediately the generalized bound of Ma Chunsheng (1992). Some further remarks with regard to extensions are made.