On the asymptotic normality of hierarchical mixtures-of-experts for generalized linear models

In the class of hierarchical mixtures-of-experts (HME) models, “experts” in the exponential family with generalized linear mean functions of the form ψ(α+x^Tβ) are mixed, according to a set of local weights called the “gating functions” depending on the predictor x. Here ψ(·) is the inverse link function. We provide regularity conditions on the experts and on the gating functions under which the maximum-likelihood method in the large sample limit produces a consistent and asymptotically normal estimator of the mean response. The regularity conditions are validated for Poisson, gamma, normal, and binomial experts