Minimax Optimal Rate for Parameter Estimation in Multivariate Deviated Models

2023/01/27 by Dat Do, Do, Dat, Huy Nguyen +5 · 1 citation
Engineering · Mathematics · #Control Systems and Identification #Statistical Methods and Inference #Advanced Statistical Methods and Models

paper · pdf · doi:10.48550/arxiv.2301.11808

Abstract

We study the maximum likelihood estimation (MLE) in the multivariate deviated model where the data are generated from the density function (1-λ^∗)h₀(x)+λ^∗f(x|μ^∗, Σ^∗) in which h₀ is a known function, λ^∗ ∈ [0,1] and (μ^∗, Σ^∗) 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 h₀ and the density function f; (2) The deviated proportion λ^∗ can go to the extreme points of [0,1] as the sample size tends to infinity. To address these challenges, we develop the distinguishability condition to capture the linear independent relation between the function h₀ and the density function f. We then provide comprehensive convergence rates of the MLE via the vanishing rate of λ^∗ to zero as well as the distinguishability of two functions h₀ and f.

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