Variation of the expectation and quantiles of the mixture of ordered distributions under imprecise choice of prior

We consider families of probability distributions indexed by the parameters from an interval in the real line. We merely assume that the increase of parameter results in the increase of corresponding distribution with respect to the standard stochastic order. We investigate consequences of imprecise choice of prior distribution on the parameters of the model. Namely, we assume a fixed prior whereas the true one belongs to a nonparametric neighbourhood of it. We determine sharp lower and upper bounds on the differences between the expectations and quantiles of the mixtures with respect to the true and assumed mixing distributions in various scale units.