A simple application of binomial—negative binomial relationship in the derivation of sharp bounds for moments of order statistics based on greatest convex minorants

In this note, we present an alternate way of obtaining the sharp bound for expected values of order statistics derived by Moriguti (1953) based on the principle of greatest convex minorants. The formula presented here using an application of binomial—negative binomial relationship, in addition to being simpler to use for numerical computations, also yields explicit bounds for a few extreme order statistics.