Bounds on Expectations of Order Statistics

Abstract

The classic results on universal bounds for order statistics were provided in papers published simultaneously by Gumbel (1954) and Hartley and David (1954). Antecedents and partial anticipations can be identified, particularly noteworthy is the contribution of Plackett (1947). These authors all dealt with the i.i.d. case. Relaxation of the identical distribution and the independence assumptions was not explicitly treated until 25 years later, though again one can identify relevant insights throughout the intervening period. Two papers which turned out to be influential in refocussing attention on variations on the Gumbel-Hartley-David theme were Samuelson (1968) and Lai and Robbins (1976). Samuelson’s note, with its irresistable title “How deviant can you be” spawned a torrent of generalizations, several of which referred to bounds on order statistics. It also spawned a flurry of rediscoveries of earlier notes on these topics. Ultimate priority seems hard to pin down although Scott’s (1936) appendix to the Pearson and Chandra Sekar paper stands out as one of the earliest sources thus far identified. Lai and Robbins (1976) introduced a class of maximally dependent joint distributions. The name maximally dependent is perhaps an infelicitous choice but apparently we are stuck with it. In any case, such joint distributions conveniently provide extreme cases for distributions of possibly dependent maxima. Sections 1 through 3 will survey the universal bounds obtainable using all the aforementioned techniques.