Simultaneous Test Of High Dimensional Population Mean Vector And Covariance Matrix Based On Modified Likelihood Ratio

With rapid developments in and extensive application of computers,the storage and analysis of high-dimensional data are applied in many fields.A common feature of all these high-dimensional data is that their dimensions can be proportionally large when compared with the sample size,such as the analysis of the stock market in economics,signal processing in wireless communications,and the study of sequences of genes in biology.In the classical multivariate statistical analysis,statisticians usually consider estimation and hypothesis tests under the assumption of a large sample size n,but the dimension p of the random vector is fixed.Consequently,the traditional estimation and hypothesis test tools in the classical multivariate statistical analysis are no more valid or perform badly,which makes the statisticians devote to developing new methods for such high-dimensional dataIn this paper,we primarily focus on simultaneous testing mean vector and covariance matrix with high-dimensional data using random matrix theory.First of all,we study the simultaneous test of high-dimensional mean value vector and covariance matrix of a single population.Applying the central limit theorem for linear spectral statistics of sample covariance matrices,we establish new modification for the likelihood ratio test,and find that this modified test converges in distribution to normal distribution,when the dimension p tends to infinity,proportionate to the sample size n under the null hypothesis.Furthermore,we conduct a simulation study to examine the performance of the test and compare it with other tests proposed in past studies.As the simulation results show,our empirical powers are clearly superior to those of other tests.Secondly,we consider the simultaneous testing of mean vector and covariance matrix of two populations.Applying the central limit theorem(CLT)for linear spectral statistics of Beta matrices,we obtain the asymptotic distribution of the modified LRT under the null hypothesis.In addition,we compare the proposed test with other previous studies through the simulation results.Simulations demonstrate that the modified LRT can be significantly more powerful than other tests.