  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.
