Likelihood Ratio Tests Of Two High-Dimensional Statistical Models

In many scientific fields,we often encounter the case that both the dimension p and the sample size n are very large,this is often called "large p,large n" in the literature.In classic multivariate analysis,the dimension p is often fixed or relatively small compared with the sample size n,The traditional likelihood ratio method can solve the low dimen-sional case effectively,but it is no longer valid for high dimensional problems,so it is very necessary to find new methods to solve the high dimensional problem,and it has very practical significance.The paper mainly studies the hypothesis testing of two models that both dimension p and sample size n are very large.One of the models is a significance test of regression variable in the high-dimensional linear regression model.The paper proposed methods to solve the problem under two slightly different cases,and compared it with three other methods of the classical likelihood ratio test method(LRT),the Box's method(BOX)and the random matrix theory method(RMT)by Bai through Matlab simulations.The simulations demonstrate that the proposed high-dimensional likelihood ratio test method(HLRT)in the paper outperforms the classical LRT method and BOX method,and performs as well as RMT method for high dimensional problems.Another model is a likelihood ratio test of intraclass correlation covariance structure for high-dimensional normal vectors.Under three different assumptions,we proved that the logarithm likelihood ratio test statistic will converge in distribution to a Gaussian random variable respectively.We compared our proposed normal approximation(HLRT)with the traditional Chi-square approximation(BOX)and F-approximation(F)finally,and the Matlab simulations show that the likelihood ratio test by using our proposed method outperforms those using the traditional methods in dealing with high dimensional data.