Identifiability of the proportion of null hypotheses in skew-mixture models for the p-value distribution

Abstract

In many multiple testing procedures, accurate modeling of the p-value distribution is a key issue. Mixture distributions have been shown to provide adequate models for p-value densities under the null and the alternative hypotheses. An important parameter of the mixture model that needs to be estimated is the proportion of true null hypotheses, which under the mixture formulation becomes the probability mass attached to the value associated with the null hypothesis. It is well known that in a general mixture model, especially when a scale parameter is present, the mixing distribution need not be identifiable. Nevertheless, under our setting for mixture model for p-values, we show that the weight attached to the null hypothesis is identifiable under two very different types of conditions. We consider several examples including univariate and multivariate mixture models for transformed p-values. Finally, we formulate an abstract theorem for general mixtures and present other examples.