  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 $f$.
