Joint mean and covariance estimation with unreplicated matrix-variate data
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It has been proposed that complex populations, such as those that arise in genomics studies, may exhibit dependencies among the observations as well as among the variables. This gives rise to the challenging problem of analyzing unreplicated high-dimensional data with unknown mean and dependence structures. Matrix-variate approaches that impose various forms of (inverse) covariance sparsity allow flexible dependence structures to be estimated, but cannot directly be applied when the mean and dependences are estimated jointly. We present a practical method utilizing penalized (inverse) covariance estimation and generalized least squares to address this challenge. The advantages of our approaches are: (i) dependence graphs and covariance structures can be estimated in the presence of unknown mean structure, (ii) the mean structure becomes more efficiently estimated when accounting for the dependence structure(s); and (iii) inferences about the mean parameters become correctly calibrated. We establish consistency and obtain rates of convergence for estimating the mean parameters and covariance matrices. We use simulation studies and analysis of genomic data from a twin study of ulcerative colitis to illustrate the statistical convergence and the performance of the procedures in practical settings. Furthermore, several lines of evidence show that the test statistics for differential gene expression produced by our method are correctly calibrated.
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