Convergence properties of functional estimates for discrete distributions
Convergence properties of functional estimates for discrete distributions
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DOI：
10.1002/rsa.10019
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发表时间：
2001-10-01
影响因子：
1
通讯作者：
Kontoyiannis, I
中科院分区：
文献类型：
--
作者：
Antos, A;Kontoyiannis, I
假设P是可数字母表x上的任意离散分布。根据身份证。样本（X-1，...，X-n），我们考虑了未知分布P的熵H（P）或其它泛函F = F（P）的估计问题，证明了对于满足较弱条件（包括均值、熵和互信息的情形）的可加泛函，F的插入估计是普适的.我们还证明，没有进一步的假设，没有收敛速度的结果可以得到任何序列的估计。在熵估计的情况下，在各种不同的假设下，我们得到了插入估计和基于匹配长度的非参数估计的收敛速度结果。插件估计的方差和预期误差的行为被证明是在形成鲜明对比的有限字母表的情况下。一些其他重要的例子泛函也处理在一些细节。(C)John Wiley & Sons，Inc.
Suppose P is an arbitrary discrete distribution on a countable alphabet x. Given an i.i.d. sample (X-1,..., X-n) drawn from P, we consider the problem of estimating the entropy H(P) or some other functional F = F(P) of the unknown distribution P. We show that, for additive functionals satisfying mild conditions (including the cases of the mean, the entropy, and mutual information), the plug-in estimates of F are universally consistent. We also prove that, without further assumptions, no rate-of-convergence results can be obtained for any sequence of estimators. In the case of entropy estimation, under a variety of different assumptions, we get rate-of-convergence results for the plug-in estimate and for a nonparametric estimator based on match-lengths. The behavior of the variance and the expected error of the plug-in estimate is shown to be in sharp contrast to the finite-alphabet case. A number of other important examples of functionals are also treated in some detail. (C) 2001 John Wiley & Sons, Inc.