Minimax Optimal Rate for Parameter Estimation in Multivariate Deviated Models

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 $f$.