A Minimax Approach to Errors-in-Variables Linear Models

Abstract

The paper considers a simple Errors-in-Variables (EiV) model Y_i = a + bX_i + εξ_i; Z_i= X_i + σζ_i, where ξ_i, ζ_i are i.i.d. standard Gaussian random variables, X_i ∈ ℝ are unknown non-random regressors, and ε, σ are known noise levels. The goal is to estimates unknown parameters a, b ∈ ℝ based on the observations {Y_i, Z_i, i = 1, …, n}. It is well known [3] that the maximum likelihood estimates of these parameters have unbounded moments. In order to construct estimates with good statistical properties, we study EiV model in the large noise regime assuming that n → ∞, but ${\epsilon ^2} = \sqrt n \epsilon _ \circ ^2,{\sigma ^2} = \sqrt n \sigma _ \circ ^2$ with some $\epsilon_\circ^2, \sigma_\circ^2>0$. Under these assumptions, a minimax approach to estimating a, b is developed. It is shown that minimax estimates are solutions to a convex optimization problem and a fast algorithm for solving it is proposed.