ABSTRACT

Best linear unbiased estimators (BLUE’s) are known to be optimal in many respects under normal assumptions. Since variance minimization doesn’t depend on normality and unbiasedness is often considered reasonable, many statisticians have felt that BLUE’s ought to preform relatively well in some generality. The result here considers the general linear model and shows that any measurable estimator that is unbiased over a moderately large family of distributions must be linear. Thus, imposing unbiasedness cannot offer any improvement over imposing linearity. The problem was suggested by Hansen, who showed that any estimator unbiased for nearly all error distributions (with finite covariance) must have a variance no smaller than that of the best linear estimator in some parametric subfamily. Specifically, the hypothesis of linearity can be dropped from the classical Gauss–Markov Theorem. This might suggest that the best unbiased estimator should provide superior performance, but the result here shows that the best unbiased regression estimator can be no better than the best linear estimator.

摘要

已知最佳线性无偏估计量 (BLUE) 在正常假设下在许多方面都是最优的。由于方差最小化不依赖于正态性并且无偏性通常被认为是合理的，因此许多统计学家认为 BLUE 应该在某些普遍性上表现得相对较好。这里的结果考虑了一般线性模型，并表明任何在中等大的分布族上无偏的可测量估计量必须是线性的。因此，强加无偏性不能比强加线性提供任何改进。Hansen 提出了这个问题，他表明任何对几乎所有误差分布（具有有限协方差）无偏的估计器的方差必须不小于某些参数子族中最佳线性估计器的方差。具体来说，线性假设可以从经典的高斯-马尔可夫定理中删除。这可能表明最好的无偏估计器应该提供卓越的性能，但这里的结果表明，最好的无偏回​​归估计器不会比最好的线性估计器更好。