Minimax Estimation of the L1 Distance
Minimax Estimation of the L1 Distance
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DOI：
10.1109/tit.2018.2846245
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发表时间：
2018-10-01
影响因子：
2.5
通讯作者：
Weissman, Tsachy
中科院分区：
文献类型：
--
作者：
Jiao, Jiantao;Han, Yanjun;Weissman, Tsachy
我们考虑了在非扰动和大型字母设置中从经验数据中估算两个离散概率度量P和Q之间L-1距离的问题。当已知Q并从P中获得n个样品时，我们表明，对于每个Q，带有N样品的最小速率 - 最佳估计器可实现与N Inn样品的最大似然估计器相当的性能。当P和Q均未清楚时，我们构建了最小速率 - 最佳估计器，其最坏情况的性能本质上是已知的Q情况，Q是均匀的，这意味着Q是均匀的，本质上是最困难的案例。 Jiao等人确定的有效样本量扩大现象在每个Q和Q未知病例的已知Q情况下都持有。但是，与（1）平行的P -Q平行的最佳估计器的构建需要超出Jiao等人的基于近似功能估计方法的新技术和见解。
We consider the problem of estimating the L-1 distance between two discrete probability measures P and Q from empirical data in a nonasymptotic and large alphabet setting. When Q is known and one obtains n samples from P, we show that for every Q, the minimax rate-optimal estimator with n samples achieves performance comparable to that of the maximum likelihood estimator with n Inn samples. When both P and Q are unknown, we construct minimax rate-optimal estimators, whose worst case performance is essentially that of the known Q case with Q being uniform, implying that Q being uniform is essentially the most difficult case. The effective sample size enlargement phenomenon, identified by Jiao et aL, holds both in the known Q case for every Q and the Q unknown case. However, the construction of optimal estimators for parallel to P - Q parallel to(1) requires new techniques and insights beyond the approximation-based method of functional estimation by Jiao et al.