On Parameter Estimation in Deviated Gaussian Mixture of Experts

We consider the parameter estimation problem in the deviated Gaussian mixture of experts in which the data are generated from (1 - λ^∗) g_0(Y| X)+ λ^∗∑_i = 1^k_∗ p_i^∗ f(Y|(a_i^∗)^⊤X+b_i^∗,σ_i^∗), where X, Y are respectively a covariate vector and a response variable, g_0(Y|X) is a known function, λ^∗∈ [0, 1] is true but unknown mixing proportion, and (p_i^∗, a_i^∗, b_i^∗, σ_i^∗) for 1 ≤ i ≤ k^∗ 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.