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.

中文翻译：

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变量误差线性模型的Minimax方法

本文考虑了简单的错误，在变量（EIV）模型ÿ_我=一个+ BX_我+ εξ_我; ž_我= X_我+ σζ_我，其中ξ_我，ζ_我是独立同分布的标准的高斯随机变量，X_我∈ℝ是未知的非随机回归量，和ε，σ是已知的噪声水平。我们的目标是估计未知参数一，b基于观测{∈ℝ ÿ_我，Z _i，i = 1，…，n }。众所周知[3]，这些参数的最大似然估计具有无穷矩。为了构建具有良好统计特性的估计，我们在大噪声条件下假设n →∞，但是\（{\ epsilon ^ 2} = \ sqrt n \ epsilon _ \ circ ^ 2，{\ sigma ^ 2} = \ sqrt n \ sigma _ \ circ ^ 2 \）和一些\（\ epsilon_ \ circ ^ 2，\ sigma_ \ circ ^ 2> 0 \）。在这些假设下，开发了一种估计a，b的极小极大方法。结果表明，最小极大估计是凸优化问题的一种解决方案，并提出了一种快速算法。