Limiting Distributions Of The Likelihood Ratio Test Statistics For Independence Of Normal Random Vectors

This paper mainly focuses on the research of the limit distribution of likelihood ratio test(LRT)statistic for the independence test of sub-vectors in p-dimensional normal distribution random vectors.In previous studies,Qi(2019)proved that under specific conditions,the limit distribution of LRT statistic is asymptotically normal.However,this paper aims to explore the limit distribution of LRT statistic under general conditions and reveal the necessary and su cient conditions for LRT statistic to converge to the normal distribution when p tends to infinity.First,we studied the limit distribution of LRT statistic under general conditions in detail and identified all possible types.Through the analysis of different scenarios,we summarized the conditions for LRT statistic to converge to the normal distribution.These conditions provide a broader theoretical basis for studying the statistical properties of high-dimensional data.Next,we discussed the limit distribution of the adjusted LRT test statistic proposed in Qi(2019).We found that this adjustment method improved the performance of the LRT statistic to some extent,making it more applicable in more general situations.To further evaluate the performance of the adjusted LRT test statistic,we conducted detailed numerical simulation experiments.In addition,we compared the performance of the classical chi-square approximation,normal approximation,and non-normal approximation for the LRT statistic through simulation results.We found that under specific conditions,the normal approximation and non-normal approximation might be more accurate than the classical chi-square approximation in some cases.However,in other cases,the chi-square approximation still has good performance.These findings are of great significance for understanding the performance of the LRT statistic under different conditions.