Confidence sets based on penalized maximum likelihood estimators in Gaussian regression
Confidence sets based on penalized maximum likelihood estimators in Gaussian regression
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DOI：
10.1214/09-ejs523
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发表时间：
2010-01-01
影响因子：
1.1
通讯作者：
Schneider, Ulrike
中科院分区：
文献类型：
--
作者：
Poetscher, Benedikt M.;Schneider, Ulrike
基于惩罚的最大似然估计，如LASSO，自适应LASSO，和硬阈值的置信区间进行了分析。在已知方差的情况下，有限样本覆盖性质，这样的间隔被确定，它表明，对称的间隔是最短的。基于硬阈值估计的最短间隔的长度大于基于自适应LASSO的最短间隔的长度，其大于基于LASSO的最短间隔的长度，其进而大于基于最大似然估计的标准间隔。在惩罚估计量被调整为具有“稀疏性”的情况下，基于这些估计量的区间比标准区间大一个数量级。此外，一个简单的渐近置信区间的建设，在“稀疏”的情况下，也适用于平滑剪切绝对偏差估计，进行了讨论。已知方差的情况下的结果进行到未知方差的情况下，在适当的渐近意义。
Confidence intervals based on penalized maximum likelihood estimators such as the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the known-variance case, the finite-sample coverage properties of such intervals are determined and it is shown that symmetric intervals are the shortest. The length of the shortest intervals based on the hard-thresholding estimator is larger than the length of the shortest interval based on the adaptive LASSO, which is larger than the length of the shortest interval based on the LASSO, which in turn is larger than the standard interval based on the maximum likelihood estimator. In the case where the penalized estimators are tuned to possess the 'sparsity property', the intervals based on these estimators are larger than the standard interval by an order of magnitude. Furthermore, a simple asymptotic confidence interval construction in the 'sparse' case, that also applies to the smoothly clipped absolute deviation estimator, is discussed. The results for the known-variance case are shown to carry over to the unknown-variance case in an appropriate asymptotic sense.