Mixtures of exponential distributions and stochastic orders (Q1613037)

Let \(X\) and \(Y\) be two random variables with respective distribution functions \(F\) and \(G\). Denote by \(\widetilde{X}\) the random variable whose distribution function \(\widetilde{F}\) is a mixture of exponential distributions given by \(\widetilde{F}(s)=\int_0^\infty(1-e^{-sx}) dF(x)\), \(s\geq 0\). Similarly define \(\widetilde{Y}\). The author finds conditions under which various stochastic order relationships between \(X\) and \(Y\) yield similar stochastic order relationships between \(\widetilde{X}\) and \(\widetilde{Y}\).