We study the maximum likelihood estimation (MLE) in the multivariate deviated model where the data are generated from the density function $(1-\lambda^{\ast})h_{0}(x)+\lambda^{\ast}f(x|\mu^{\ast}, \Sigma^{\ast})$ in which $h_{0}$ is a known function, $\lambda^{\ast} \in [0,1]$ and $(\mu^{\ast}, \Sigma^{\ast})$ 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 $\lambda^{\ast}$ can go to the extreme points of $[0,1]$ as the sample size tends to infinity. To address these challenges, we develop the \emph{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 $\lambda^{\ast}$ to zero as well as the distinguishability of two functions $h_{0}$ and $f$.

翻译：我们研究多元偏态模型中的最大似然估计（MLE），其中数据由密度函数$(1-\lambda^{\ast})h_{0}(x)+\lambda^{\ast}f(x|\mu^{\ast}, \Sigma^{\ast})$生成，$h_{0}$为己知函数，$\lambda^{\ast} \in [0,1]$和$(\mu^{\ast}, \Sigma^{\ast})$为待估计的未知参数。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）随着样本量趋于无穷，偏态比例$\lambda^{\ast}$可能趋近于$[0,1]$的极端点。为应对这些挑战，我们提出了\emph{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。进而，我们基于$\lambda^{\ast}$趋近于零的速率以及两个函数$h_{0}$和$f$的可区分性，给出了MLE的全面收敛速率。