Central Limit Theorems For Classical Likelihood Ratio Tests For High-dimensional Normal Distributions

Select a random sample of sizen from p variatenormal distributions.In many classical textbooks on multivariate analysis, there are many testsfor their means and covariance matrices in high-dimensional setting. Suchas Muirhead（1982）[8], Eaton（1983）[4]and Anderson（1958）[1]. Many traditionalstatistical theories have been largely used, which made great values. Eventhough they are great, they also have their shortages. Traditionaltheories are in the setting that the size of the sample is large whichcan go to infinity, that the dimension of the dataset, denoted by p, isconsidered as a fixed small constant or at least negligible compared withthe sample size n.Along with the progress of science and technology, the rapiddevelopment of computer technology, the datum that we can get become moreand more, the dimension of the datum become bigger and bigger. So thatwe cannot neglect. Face to these datum, the efficiency of traditionaltheories will be gradually reduced with the increase of the dimension.In this paper, my work is studying the case that under the assumptionthat the sample size n is also large to infinity, but the dimension ofthe dataset is not a fixed constant but also a large number can beproportionally large compared with the sample size n.We can give sample size n and dimension p like that:nâ†'∞, pâ†'∞且p/nâ†'y∈[0,1].Under this assumption, we consider about the test that:H0: Σ=I p vs H1:Σ≠Ip.We care about its likelihood ratio test statistics and the centrallimit theory for those statistics. Here is the layout of this paper: in my first section, I will givethe original and development of the question, the prepare knowledge. Thesecond section, this is the major part of this paper, it has been dividedinto three parts. In the first part, introduce the test:H0: Σ=I p vs H1:Σ≠Ip, give two traditional test statistics, give thecentral limit theory of the two traditional test statistics. In the secondpart, I will prove the above theories. In the last part, I will give thesimulation to make a comparison between traditionalχ2approximatedistribution and CLT. So that we can easily see the advantage of the CLTin the high dimensional setting. In my third section, I will give aconclusion and give a prospect of study on high dimensional data.