We consider the problem of adaptive estimation of a linear functional of an unknown multivariate vector from its observations against white Gaussian noise. As a family of estimators for the functional, we use those generated by projection estimators of the unknown vector, and the main problem is to select the best estimator in this family. The goal of the paper is to explain and mathematically justify a simple statistical idea used in adaptive (i.e., observation-based) choice of the best estimator of a linear functional from a given family of estimators. We also discuss generalizations of the considered statistical model and the proposed estimation method, which allow to cover a broad class of statistical problems.

中文翻译：

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从对白噪声的观察来看线性函数的自适应估计

我们考虑从未知多变量矢量对白高斯噪声的观察中自适应估计线性函数的问题。作为函数的估计量族，我们使用未知向量的投影估计量生成的估计量，主要问题是选择该族中的最佳估计量。本文的目的是解释和数学证明一种简单的统计思想，该思想用于从给定的估计量族中自适应选择（即基于观测）线性函数的最佳估计量。我们还讨论了所考虑的统计模型和所提出的估计方法的概括，它们可以涵盖广泛的统计问题。