Abstract Inference in a high-dimensional situation may involve regularization of a certain form to treat overparameterization, imposing challenges to inference. The common practice of inference uses either a regularized model, as in inference after model selection, or bias-reduction known as “debias.” While the first ignores statistical uncertainty inherent in regularization, the second reduces the bias inbred in regularization at the expense of increased variance. In this article, we propose a constrained maximum likelihood method for hypothesis testing involving unspecific nuisance parameters, with a focus of alleviating the impact of regularization on inference. Particularly, for general composite hypotheses, we unregularize hypothesized parameters whereas regularizing nuisance parameters through a L0-constraint controlling the degree of sparseness. This approach is analogous to semiparametric likelihood inference in a high-dimensional situation. On this ground, for the Gaussian graphical model and linear regression, we derive conditions under which the asymptotic distribution of the constrained likelihood ratio is established, permitting parameter dimension increasing with the sample size. Interestingly, the corresponding limiting distribution is the chi-square or normal, depending on if the co-dimension of a test is finite or increases with the sample size, leading to asymptotic similar tests. This goes beyond the classical Wilks phenomenon. Numerically, we demonstrate that the proposed method performs well against it competitors in various scenarios. Finally, we apply the proposed method to infer linkages in brain network analysis based on MRI data, to contrast Alzheimer’s disease patients against healthy subjects. Supplementary materials for this article are available online.

中文翻译：

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关于高维约束最大似然推理*

摘要 高维情况下的推理可能涉及某种形式的正则化以处理过参数化，给推理带来挑战。推理的常见做法要么使用正则化模型，如模型选择后的推理，要么使用称为“debias”的减少偏差。虽然第一个忽略了正则化中固有的统计不确定性，但第二个以增加方差为代价减少了正则化中的自交偏差。在本文中，我们提出了一种约束最大似然方法，用于涉及非特定干扰参数的假设检验，重点是减轻正则化对推理的影响。特别是对于一般复合假设，我们不规范化假设参数，同时通过控制稀疏程度的 L0 约束规范化干扰参数。这种方法类似于高维情况下的半参数似然推理。在此基础上，对于高斯图模型和线性回归，我们推导了建立约束似然比渐近分布的条件，允许参数维数随着样本量的增加而增加。有趣的是，相应的极限分布是卡方分布或正态分布，这取决于检验的共维是有限的还是随着样本量的增加而增加，从而导致渐近的相似检验。这超出了经典的威尔克斯现象。数字上，我们证明了所提出的方法在各种情况下与竞争对手相比表现良好。最后，我们应用所提出的方法来推断基于 MRI 数据的脑网络分析中的联系，以将阿尔茨海默病患者与健康受试者进行对比。本文的补充材料可在线获取。