辅助信息的最佳利用：信息几何与经验过程
Optimal use of auxiliary information : information geometry and empirical process
摘要：
我们把有限期望集合中的辅助信息以最优信息几何的方式引入到经验测度中。这允许统一利用边信息的几种方法，并唯一地定义知情的经验测量。这些方法被证明共享相同的渐近性质。然后研究了真实信息下的知情经验过程。在最小假设条件下，建立了经验测度的Glivenko-Cantelli和Donsker定理，并量化了渐近一致方差缩减.此外，我们证明了知情的经验过程比经典的经验过程更集中的所有大$n$。最后，作为方差减少的一个例子，我们将其中一些结果应用于知情的经验分位数。
英文摘要：
We incorporate into the empirical measure the auxiliary information given by a finite collection of expectation in an optimal information geometry way. This allows to unify several methods exploiting a side information and to uniquely define an informed empirical measure. These methods are shown to share the same asymptotic properties. Then we study the informed empirical process subject to a true information. We establish the Glivenko-Cantelli and Donsker theorems for the informed empirical measure under minimal assumptions and we quantify the asymptotic uniform variance reduction. Moreover, we prove that the informed empirical process is more concentrated than the classical empirical process for all large $n$. Finally, as an illustration of the variance reduction, we apply some of these results to the informed empirical quantiles.
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