On High-Dimensional Constrained Maximum Likelihood Inference

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.