Lower bounds for the stability of normal mixture models with respect to perturbations of mixing distribution

The stability of normal mixture models with respect to perturbations of mixing distribution is investigated. Inequality estimating the distance between two mixing distributions through the closeness of the corresponding mixtures is presented. Existence theorem for stability estimates is proved for subclasses of scale and shift mixtures of normal distributions. For the class of shift mixtures, the estimate is obtained in an explicit form. It is shown that the presented results cannot be radically improved without additional assumptions.