Likelihood Ratio Tests Of High-dimensional Statistical Models

In the Age of Big Data, we always encounter a lot of high-dimensional problems and all of these problems have the feature that both the dimension p and the sample size n are very large, this is often called "large p, large n". The traditional multivariate statistics analysis can resolve the problem that its dimension p is small or fixed very well such as classical Chi-square approximation method and likelihood ratio test method and so on. But with the increases of the dimension p, all of these methods can't deal with the problems well and even failure. So it is a very meaningful work to find some new methods to solve the high-dimensional problems.This article is to consider the high-dimensional hypothesis test of two models that both the dimension p and the sample size n are very large. At first, we study the high-dimensional hypothesis test on circular symmetric covariance structure. Under two slight-ly different cases, by using the continuity theorem of moment generating function and the asymptotic expansion of ?-function, we prove that under the assumption of normality,the likelihood ratio test statistic converges in distribution to a normal distributed ran-dom variable. The simulations indicate that our high-dimensional likelihood ratio test method (HLRT) is better than that using Chi-square approximation method (BOX) or high-dimensional edgeworth expansion method (HEE), and it is as effective as the more accurate high-dimensional edgeworth expansion method (AHEE) on analyzing the high-dimensional data.The chapter three deals with the likelihood ratio test for equality of the smallest eigenvalues in high-dimensional principal component analysis. Under null hypothesis and the assumption of normality, by using the continuity theorem of characteristic function and the similar expansion method, we derive that the logarithm likelihood ratio test statistic subject to normal distribution. Numerical simulations reveal that our proposed normal approximation method (HLRT) performs as well as the more accurate high-dimensional asymptotic expansion method (AHAE), and all of them are more accurate than the Chi-square approximation method (Lawley) when dimension p increases.