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We consider hierarchical mixtures-of-experts (HME) models where exponential family regression models with generalized linear mean functions of the form ψ(α + x T β) are mixed. Here ψ(.) is the inverse link function. Suppose the true response y follows an exponential family regression model with mean function belonging to a class of smooth functions of the form ψ(h(x)) where h(.)∈ W∞ 2; K0 (a Sobolev class over [0, 1] s ). It is shown that the HME probability density functions can approximate the true density, at a rate of O(m -2/s ) in Hellinger distance and at a rate of O(m -4/s ) in Kullback-Leibler divergence, where m is the number of experts, and s is the dimension of the predictor x. We also provide conditions under which the mean-square error of the estimated mean response obtained from the maximum likelihood method converges to zero, as the sample size and the number of experts both increase.

Hierarchical mixtures-of-experts for exponential family regression models: approximation and maximum likelihood estimation