Statistical Estimation of the Shannon Entropy  

作　　者：Alexander BULINSKI Denis DIMITROV 

机构地区：[1]Steklov Mathematical Institute of Russian Academy of Sciences and Department of Mathematics and Mechanics,Lomonosov Moscow State University,Moscow 119234,Russia

出　　处：《Acta Mathematica Sinica,English Series》2019年第1期17-46,共30页数学学报（英文版）

基　　金：Supported by the Russian Science Foundation(Grant No.14-21-00162)

摘　　要：The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.The behavior of the Kozachenko–Leonenko estimates for the(differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy–Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.

关 键 词：Shannon differential entropy Kozachenko-Leonenko estimates HARDY-LITTLEWOOD maxi-mal function ANALOGUES ASYMPTOTIC UNBIASEDNESS and L^2-consistency Gaussian VECTORS 

分 类 号：O1[理学—数学]