On the Increasing Convex Order of Relative Spacings of Order Statistics

Abstract

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions.

Some stochastic comparisons of lower records and lower record spacings

“…Now, upon substituting ( 19) and ( 20) in (17) and changing the order of integration by Fubini's theorem, we see that ( 12) is equivalent to 𝑚≥ 𝑗 𝑟 ≥𝑚 𝑚𝑃(𝐿 𝑗 = 𝑚)𝐼 (𝑠, 𝑟) ≥ 0, for all 𝑠. (21) As 𝑗 ≥ 𝑖, it is clear that ( 16) implies (21), which proves the required result for the increasing convex order.…”

“…Some other works dealing with stochastic comparisons of spacings of GOS and records are due to Hu and Zhuang [24], Belzunce et al [16,18] , Zhao et al [38] and Zhuang and Hu [39]. Stochastic comparisons of relative spacings have also been discussed in the literature by Belzunce et al [19] and Castaño-Martínez et al [21]. Applications of stochastic comparisons of minimum order statistics in welfare theory have been illustrated recently by Castaño-Martínez et al [20].…”