High-dimensional graphical inference via partially penalised regression

Graphical models are important tools to characterise the conditional independence structure among a set of variables. Despite the rapid development of statistical inference for high-dimensional graphical models, existing methods typically need a stringent constraint on the sample size. In this paper, we develop a new graphical projection estimator (GPE) for statistical inference in Gaussian graphical models via partially penalised regression. The suggested inference procedure takes advantage of the strong signals, which can be identified in advance, and utilises partially penalised regression to avoid the penalisation on them when constructing the GPE. It leads to enhanced inference efficiency by removing the impacts of strong signals that contribute to the bias term. We show that the proposed GPE can enjoy asymptotic normality under a relaxed constraint on the sample size, which is of the same order as that needed for consistent estimation. The usefulness of our method is demonstrated through simulations and a prostate tumour gene expression dataset.