Dispersive ordering and the total time on test transformation (Q1091063)

Let \({\mathcal F}\) be the class of absolutely continuous distributions F such that \(F(0)=0\) with positive and right- or (left-) continuous density on the interval where \(0<F(x)<1\). Denote by \(F^{-1}\) the inverse of F. The function \(F\in {\mathcal F}\) is said to be dispersed with respect to \(G\in {\mathcal F}\) if \(F^{-1}(\beta)-F^{-1}(\alpha)\leq G^{-1}(\beta)-G^{- 1}(\alpha)\), whenever \(0<\alpha <\beta <1.\) The author considers the problem of testing the hypothesis \(F=G\) against the alternative that F is dispersed with respect to G, \(G\in {\mathcal F}\). The decision procedure is based on the properties of the empirical analogue of the so-called total time on test transformation \[ I(F_ n)=\int^{F_ n^{-1}(t)}_{0}g(G^{-1}(F_ n(u)))du \] where \(F_ n(\cdot)\) is the empirical distribution function corresponding to F.