Testing for dispersive ordering (Q1113584)

An asymptotically distribution-free test is proposed and studied for testing the null hypothesis \(H_ 0: F=^{disp}G\) versus \(H_ 1: F<^{disp}G\), that is \[ G^{-1}(\beta)-G^{-1}(\alpha)\geq F^{- 1}(\beta)-F^{-1}(\alpha)\quad for\quad 0\leq \alpha <\beta \leq 1. \] The proposed test is based on an estimator of \(\int f^ 2(x)dz-\int g^ 2(x)dx\), where f(g) is the probability density function corresponding to the distribution function F(G). The cases of two independent samples as well as that of paired samples are discussed in detail.