The Annals of Statistics

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

Wenxin Jiang and Martin A. Tanner

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Abstract

We consider hierarchical mixtures-of-experts (HME) models where exponential family regression models with generalized linear mean functions of the form $\psi(\alpha + \mathbf{x}^T \mathbf{\beta})$ are mixed. Here $\psi(\cdot)$ 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 $\psi(h(\mathbf{x}))$ where $h(\cdot)\in W_{2; K_0}^{\infty}$ (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.

Article information

Source
Ann. Statist., Volume 27, Number 3 (1999), 987-1011.

Dates
First available in Project Euclid: 5 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1018031265

Digital Object Identifier
doi:10.1214/aos/1018031265

Mathematical Reviews number (MathSciNet)
MR1724038

Zentralblatt MATH identifier
0957.62032

Subjects
Primary: 62G07: Density estimation
Secondary: 41A25: Rate of convergence, degree of approximation

Keywords
Approximation rate exponential family generalized linear models Hellinger distance Hierarchical mixtures-of-experts Kullback-Leibler divergence maximum likelihood estimation mean square error

Citation

Jiang, Wenxin; Tanner, Martin A. Hierarchical mixtures-of-experts for exponential family regression models: approximation and maximum likelihood estimation. Ann. Statist. 27 (1999), no. 3, 987--1011. doi:10.1214/aos/1018031265. https://projecteuclid.org/euclid.aos/1018031265


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