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：我们考虑偏离高斯专家混合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。这里$X$和$Y$分别是协变量向量和响应变量，$g_{0}(Y|X)$是已知函数，$\lambda^{\ast} \in [0, 1]$是真实但未知的混合比例，而$(p_{i}^{\ast}, a_{i}^{\ast}, b_{i}^{\ast}, \sigma_{i}^{\ast})$（其中$1 \leq i \leq k^{\ast}$）是高斯专家混合模型的未知参数。该问题源于拟合优度检验，即当我们需要检验数据是否由$g_{0}(Y|X)$生成（零假设）或由整个混合模型生成（备择假设）时。基于专家函数的代数结构以及$g_0$与混合部分之间的可区分性，我们构建了新颖的基于Voronoi的损失函数来刻画模型极大似然估计（MLE）的收敛速率。我们进一步证明，与常用于高斯专家混合模型参数估计的广义Wasserstein损失函数相比，我们提出的损失函数能更精确地表征参数估计的局部收敛速率。