Minimax Optimal Rate for Parameter Estimation in Multivariate Deviated Models

TL;DR

This paper establishes the optimal convergence rates for maximum likelihood estimation in multivariate deviated models, addressing challenges from the interaction of known and unknown components and the boundary behavior of the deviated proportion.

Contribution

We introduce the distinguishability condition to analyze the MLE's convergence rates in complex multivariate deviated models with unknown parameters.

Findings

Derived the minimax optimal convergence rates for MLE

Analyzed the impact of the deviated proportion approaching zero

Provided conditions for distinguishability between functions

Abstract

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 $f$.