We study the maximum likelihood estimation (MLE) in the matrix-variate deviated models where the data are generated from the density function $(1-\lambda^{*})h_{0}(x)+\lambda^{*}f(x|\mu^{*}, \Sigma^{*})$ where $h_{0}$ is a known function, $\lambda^{*} \in [0,1]$ and $(\mu^{*}, \Sigma^{*})$ are unknown parameters to estimate. The main challenges in deriving the convergence rate of the MLE mainly come from two issues: (1) The interaction between the function $h_{0}$ and the density function $f$; (2) The deviated proportion $\lambda^{*}$ can go to the extreme points of $[0,1]$ as the sample size goes to infinity. To address these challenges, we develop the distinguishability condition to capture the linear independent relation between the function $h_{0}$ and the density function $f$. We then provide comprehensive convergence rates of the MLE via the vanishing rate of $\lambda^{*}$ to 0 as well as the distinguishability of $h_{0}$ and $f$.

翻译：我们研究矩阵变量偏离模型中的最大似然估计，其中数据由密度函数$(1-\lambda^{*})h_{0}(x)+\lambda^{*}f(x|\mu^{*}, \Sigma^{*})$生成，这里$h_{0}$是已知函数，$\lambda^{*} \in [0,1]$和$(\mu^{*}, \Sigma^{*})$是需要估计的未知参数。推导MLE收敛速率的主要挑战来自两个问题：（1）函数$h_{0}$与密度函数$f$之间的相互作用；（2）偏离比例$\lambda^{*}$可能随着样本量趋于无穷而趋向$[0,1]$的极端点。为应对这些挑战，我们提出了可区分性条件来刻画函数$h_{0}$与密度函数$f$之间的线性独立关系。随后，我们基于$\lambda^{*}$趋于0的速率以及$h_{0}$与$f$的可区分性，给出了MLE的全面收敛速率。