ADAPTIVE ESTIMATORS OF A MEAN MATRIX: TOTAL LEAST SQUARES VERSUS TOTAL SHRINKAGE
均值矩阵的自适应估计：总体最小二乘与总体收缩AI 解读 ↓
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均值矩阵的自适应估计：总体最小二乘与总体收缩
- 研究问题
- 对带加性随机误差的未知均值矩阵，如何自适应平衡偏差与方差以降低二次风险，并在列间具有线性关系时比较不同估计方法？
- 方法与思路
- 作者在若干线性估计量类中最小化估计风险，构造自适应总体最小二乘和总体收缩估计，并采用行数增大、列数固定的渐近框架。
- 材料陈述的贡献
- 摘要称总体最小二乘达到降秩拟合类中的最小渐近风险，而总体收缩风险更低；还建立其与 Efron–Morris 估计的联系及秩约束比较结果。
- 阅读关注点
- 应核对随机误差条件、估计风险的构造、各比较类与秩约束，以及信噪比和误差方差有界时风险均匀收敛的具体含义。
- 阅读边界
- 仅依据所给 RePEc 出版社摘要版本，风险优势限于指定渐近框架和比较类；摘要未提供有限样本改善幅度或更一般维度增长结论。
原文摘要
An unknown constant matrix M is observed with additive random error. The basic problem considered is to devise an estimator of M that trades off bias against variance so as to achieve relatively low quadratic risk. This paper develops an adaptive total least squares estimator and an adaptive total shrinkage estimator of M that minimize estimated risk over certain large classes of linear estimators. It is shown that the asymptotic risk of the adaptive total least squares estimator is the smallest attainable among reduced rank total least squares fits to the data matrix. The asymptotic risk of the adaptive total shrinkage estimator is shown to be smaller still. A close link is established between total shrinkage and the Efron–Morris estimator of M. In the asymptotics, the row dimension of M tends to infinity, and the column dimension stays fixed. The risks converge uniformly when the signal-to-noise ratio and the measurement error variance are both bounded. A second problem treated is estimation of M under the assumption that a linear relation holds among its columns. In this formulation of the errors-in-variables linear regression model, rank constrained adaptive total least squares asymptotically dominates the usual total least squares estimator of M, and rank constrained adaptive total shrinkage is better still.This research was supported in part by National Science Foundation Grant DMS 0404547.