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 .