Moderate Deviation Principles For Classical Likelihood Ratio Tests Of High-dimensional Normal Distributions

For random samples of size n obtained from p-variate normal distributions,we con-sider the classical likelihood ratio tests for their means and covariance matrices in the high-dimensional setting.These test statistics have been extensively studied as n goes infinity and p remains fixed.In 2017,Jiang and Wang studied the moderate deviation principle of the likelihood ratio test,their results give the exponential convergence rate of the LRT statistic to the corresponding asymptotic distribution.Let x1,...,xn be a random sample from a Gaussian random vector of dimension p<n with mean vector ? and covariance matrix ?.We assume that both the dimension p and sample size n go to infinity in such a way that p/n?y?(0,1].Based on this sample,consider the testing problemwhere ? is an unknown constant.Jiang and Wang used Gartner-Eillis theorem to study the moderate deviation principle of likelihood ratio test,and gave a detailed proof.At the end of the paper,they gave the moderate deviation principle of likelihood ratio test under other test problems(The results can be seen in §1.3).In this paper,we mainly generalize the conclusions of Jiang and Wang,and study the moderate deviation principle of likelihood ratio test under six test problems in high-dimensional setting:p/n?y?[0,1).Under HO in(1).Theorem 2.1 If(?)satisfies the moderate de-viation principle with speed an2 and good rate function I(x)=x2/2 for all x?R,where{an:n?1} is any sequence of positive numbers satisfyingi.e.,for any fixed x? 0,we have thatwhere Vn,1 is the likelihood ratio statistic of(1),#12 and#12The detailed proof of the above theorem and other theorems and proofs will be given later.