CONVERGENCE RATE OF SIEVE ESTIMATES
CONVERGENCE RATE OF SIEVE ESTIMATES
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DOI：
10.1214/aos/1176325486
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发表时间：
1994-06-01
影响因子：
4.5
通讯作者：
WONG, WH
中科院分区：
文献类型：
--
作者：
SHEN, XT;WONG, WH
在本文中，我们针对筛估计、最大似然估计（MLE）以及通过在一般参数空间中优化某些经验准则而获得的相关估计的收敛速度发展了一种一般性理论。在许多情况下，尤其是当参数空间是无穷维时，在整个参数空间上进行最大化是不可取的。在这种情况下，人们必须在原始参数空间的一个逼近空间（筛）上进行最大化，并允许逼近空间的大小随着样本量的增加而增长。这种方法被称为筛法。在最大似然估计的情况下，基于筛的最大似然估计被称为筛最大似然估计。我们发现筛估计的收敛速度由（a）准则差异的局部期望值、方差和L(2)熵以及（b）筛的逼近误差所决定。作为该理论的例证，我们讨论了一个稳健的非参数回归问题、一个混合问题和一个非参数回归问题。我们还发现，当基础空间过大时，基于在整个参数空间上进行优化的估计可能无法达到最佳可能的收敛速度，而筛估计通常不会受到这种困难的影响。
In this paper, we develop a general theory for the convergence rate of sieve estimates, maximum likelihood estimates (MLE's) and related estimates obtained by optimizing certain empirical criteria in general parameter spaces. In many cases, especially when the parameter space is infinite dimensional, maximization over the whole parameter space is undesirable. In such cases, one has to perform maximization over an approximating space (sieve) of the original parameter space and allow the size of the approximating space to grow as the sample size increases. This method is called the method of sieves. In the case of the maximum likelihood estimation, an MLE based on a sieve is called a sieve MLE. We found that the convergence rate of a sieve estimate is governed by (a) the local expected values, Variances and L(2) entropy of the criterion differences and (b) the approximation error of the sieve. A robust nonparametric regression problem, a mixture problem and a nonparametric regression problem are discussed as illustrations of the theory. We also found that when the underlying space is too large, the estimate based on optimizing over the whole parameter space may not achieve the best possible rates of convergence, whereas the sieve estimate typically does not suffer from this difficulty.