Graphical Constrained Projection Inference Approach For High-dimensional Precision Matrix

Due to the highly developed technologies,high-dimensional statistical methods have been widely applied in various domains including biology,economics,medicine,and so on.Among them,high-dimensional precision matrix,as a very powerful for-malism to measure conditional dependence structure in graphical models,has attracted more and more attention.Despite the fast growing literature,how to develop scalable and efficient statistical inference methods for precision matrix still remains unclear in high dimensions.In this paper,we develop a new method called graphical constrained projection inference(Graphical Constrained Projection Inference,GCPI)to test indi-vidual entry of the precision matrix in a scalable and efficient way.The proposed test statistics are based on the constrained projection space yielded by certain screening procedures,which combine the strengths of constrained projection and screening proce-dures,thus enjoying the scalability and the tuning free property inherited from the above two methods.Theoretically,we prove that the new inference method enjoys asymptotic normality and achieves exact inference with probability tending to one,showing gen-erally better finite sample performance.Furthermore,the proposed method is shown to achieve a tradeoff between the type I error and the average power,suggesting appealing guaranteed reliability.Compared with existing inference methods,numerical results confirm the theoretical results of our method.Besides,based on the actual sales data of Babyonline company,we combine our method and several popular machine learning methods to build forecast models.The results show that the joint models can greatly improve the prediction accuracy,which is advantageous for the company to make ar-rangements for production and management in advance.