论文标题
普通平均模型的稀疏置信度设置
Sparse Confidence Sets for Normal Mean Models
论文作者
论文摘要
在本文中，我们提出了一个新的框架，以在正常平均值$ x \ sim n（θ，σ^2i）$下构建$ d $二维稀疏参数$θ$的置信集。提议的置信度集的一个关键功能是其能力说明$θ$的稀疏性，因此称为{\ em稀疏}置信度集。这与经典方法形成鲜明对比，例如Bonferroni置信区间和其他基于重新采样的程序，在这种过程中，通常会忽略$θ$的稀疏性。具体而言，我们需要所需的稀疏置信度以满足以下两个条件：（i）均匀地在参数空间上，$θ$的覆盖范围概率高于预先指定的水平； （ii）存在$ \ {1，...，d \} $的随机子集$ s $，因此$ s $保证用于检测非零$θ_j$的预指定的真实负率（TNR）。为了利用$θ$的稀疏性，我们定义$θ_j$的置信区间将对任何$ j \ notin s $删除为单点0。在这个新框架下，我们首先考虑是否存在满足上述两个条件的稀疏信心集。为了解决这个问题，我们在适当的一类稀疏置信度集中建立了非覆盖概率的非反应最小值下限。下限解剖是稀疏性和最小信噪比（SNR）在稀疏置信集构建中的作用。此外，在SNR的适当条件下，提出了两阶段的程序来构建一个稀疏的置信度集。为了评估最佳性，显示出提出的稀疏置信度集可以达到某些正确定义的风险功能的最小较低限制，直至恒定因素。最后，我们为未知的稀疏性和SNR制定了一种自适应程序。进行数值研究以验证理论结果。
In this paper, we propose a new framework to construct confidence sets for a $d$-dimensional unknown sparse parameter $θ$ under the normal mean model $X\sim N(θ,σ^2I)$. A key feature of the proposed confidence set is its capability to account for the sparsity of $θ$, thus named as {\em sparse} confidence set. This is in sharp contrast with the classical methods, such as Bonferroni confidence intervals and other resampling based procedures, where the sparsity of $θ$ is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniformly over the parameter space, the coverage probability for $θ$ is above a pre-specified level; (ii) there exists a random subset $S$ of $\{1,...,d\}$ such that $S$ guarantees the pre-specified true negative rate (TNR) for detecting nonzero $θ_j$'s. To exploit the sparsity of $θ$, we define that the confidence interval for $θ_j$ degenerates to a single point 0 for any $j\notin S$. Under this new framework, we first consider whether there exist sparse confidence sets that satisfy the above two conditions. To address this question, we establish a non-asymptotic minimax lower bound for the non-coverage probability over a suitable class of sparse confidence sets. The lower bound deciphers the role of sparsity and minimum signal-to-noise ratio (SNR) in the construction of sparse confidence sets. Furthermore, under suitable conditions on the SNR, a two-stage procedure is proposed to construct a sparse confidence set. To evaluate the optimality, the proposed sparse confidence set is shown to attain a minimax lower bound of some properly defined risk function up to a constant factor. Finally, we develop an adaptive procedure to the unknown sparsity and SNR. Numerical studies are conducted to verify the theoretical results.
