Estimator selection in the Gaussian setting
Estimator selection in the Gaussian setting
复制标题
高斯设置中的估计器选择
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
S. Huet
中科院分区：
文献类型：
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作者：
Y. Baraud;C. Giraud;S. Huet
我们考虑具有共同未知方差独立分量的高斯向量$Y$的均值$f$的估计问题。我们的估计过程是基于估计者的选择。更准确地说，我们从基于$Y$的$f$的估计量的任意且可能无限的集合$Ff$开始，并且使用相同的数据$Y$，目标是在具有最小欧几里得风险的$Ff$中选择一个估计量。没有对估计值作出任何假设，它们相对于$Y$的相关性可能是未知的。我们建立了所选估计的非渐近风险界。作为特殊情况，我们的方法允许处理聚集和模型选择的问题，以及选择用于估计回归函数的窗口和核的问题，或者调整惩罚标准中涉及的参数的问题。当$Ff$由线性估计组成时，我们还得到了Oracle类型的不等式。为了说明这一点，我们进行了两个仿真研究。一个目的是将我们的过程与选择调谐参数的交叉验证进行比较。另一个例子是如何将我们的方法应用于解决实际中的变量选择问题。
We consider the problem of estimating the mean $f$ of a Gaussian vector $Y$ with independent components of common unknown variance $sigma^{2}$. Our estimation procedure is based on estimator selection. More precisely, we start with an arbitrary and possibly infinite collection $FF$ of estimators of $f$ based on $Y$ and, with the same data $Y$, aim at selecting an estimator among $FF$ with the smallest Euclidean risk. No assumptions on the estimators are made and their dependencies with respect to $Y$ may be unknown. We establish a non-asymptotic risk bound for the selected estimator. As particular cases, our approach allows to handle the problems of aggregation and model selection as well as those of choosing a window and a kernel for estimating a regression function, or tuning the parameter involved in a penalized criterion. We also derive oracle-type inequalities when $FF$ consists of linear estimators. For illustration, we carry out two simulation studies. One aims at comparing our procedure to cross-validation for choosing a tuning parameter. The other shows how to implement our approach to solve the problem of variable selection in practice.