We consider a linear regression model with orthogonal regressors and Gaussian random errors with known variance, in the low-dimensional setting that the length of the regression parameter vector does not exceed the length of the response vector. We suppose that we have uncertain prior information that sparsity holds, that is, that many of the components of the regression parameter vector are zero. Our aim is to construct confidence intervals for the components of this vector with both the desired coverage probability and attractive expected length properties, particularly when sparsity holds. It is known that for fixed-width confidence intervals centered on penalized maximum likelihood estimators, such as hard-thresholding, LASSO and SCAD estimators, the achievement of the desired minimum coverage necessarily results in very unattractive expected length properties. We present confidence intervals, with data-based widths, which achieve our aim. These confidence intervals dominate the usual confidence intervals with the same coverage probability, whenever the degree of sparsity exceeds a known value.

中文翻译：

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利用稀疏性的置信区间

在回归参数向量的长度不超过响应向量的长度的低维设置中，我们考虑一个具有正交回归量和已知方差的高斯随机误差的线性回归模型。我们假设我们有不确定的稀疏性先验信息，即回归参数向量的许多分量为零。我们的目标是为这个向量的分量构建置信区间，同时具有所需的覆盖概率和有吸引力的预期长度属性，特别是在稀疏性成立的情况下。众所周知，对于以惩罚最大似然估计量为中心的固定宽度置信区间，例如硬阈值、LASSO 和 SCAD 估计量，实现所需的最小覆盖范围必然导致非常不吸引人的预期长度特性。我们提出了具有基于数据的宽度的置信区间，从而实现了我们的目标。只要稀疏度超过已知值，这些置信区间就会以相同的覆盖概率支配通常的置信区间。