We consider the parameter estimation problem in the deviated Gaussian mixture of experts in which the data are generated from $(1 - \lambda^{\ast}) g_0(Y| X)+ \lambda^{\ast} \sum_{i = 1}^{k_{\ast}} p_{i}^{\ast} f(Y|(a_{i}^{\ast})^{\top}X+b_i^{\ast},\sigma_{i}^{\ast})$, where $X, Y$ are respectively a covariate vector and a response variable, $g_{0}(Y|X)$ is a known function, $\lambda^{\ast} \in [0, 1]$ is true but unknown mixing proportion, and $(p_{i}^{\ast}, a_{i}^{\ast}, b_{i}^{\ast}, \sigma_{i}^{\ast})$ for $1 \leq i \leq k^{\ast}$ are unknown parameters of the Gaussian mixture of experts. This problem arises from the goodness-of-fit test when we would like to test whether the data are generated from $g_{0}(Y|X)$ (null hypothesis) or they are generated from the whole mixture (alternative hypothesis). Based on the algebraic structure of the expert functions and the distinguishability between $g_0$ and the mixture part, we construct novel Voronoi-based loss functions to capture the convergence rates of maximum likelihood estimation (MLE) for our models. We further demonstrate that our proposed loss functions characterize the local convergence rates of parameter estimation more accurately than the generalized Wasserstein, a loss function being commonly used for estimating parameters in the Gaussian mixture of experts.

翻译：摘要：本文研究偏差高斯混合专家模型中的参数估计问题，其中数据由 $(1 - \lambda^{\ast}) g_0(Y| X)+ \lambda^{\ast} \sum_{i = 1}^{k_{\ast}} p_{i}^{\ast} f(Y|(a_{i}^{\ast})^{\top}X+b_i^{\ast},\sigma_{i}^{\ast})$ 生成，$X$ 和 $Y$ 分别表示协变量向量与响应变量，$g_{0}(Y|X)$ 为已知函数，$\lambda^{\ast} \in [0, 1]$ 为真实但未知的混合比例，$(p_{i}^{\ast}, a_{i}^{\ast}, b_{i}^{\ast}, \sigma_{i}^{\ast})$（$1 \leq i \leq k^{\ast}$）为高斯混合专家模型的未知参数。该问题源于拟合优度检验：我们需要检验数据是否由 $g_{0}(Y|X)$（原假设）生成，还是由整个混合模型（备择假设）生成。基于专家函数的代数结构以及 $g_0$ 与混合部分的可区分性，我们构建了新型的基于Voronoi图的损失函数，以表征模型最大似然估计（MLE）的收敛速率。我们进一步证明，与高斯混合专家模型参数估计中常用的广义Wasserstein损失函数相比，所提出的损失函数能更准确地刻画参数估计的局部收敛速率。