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Adjusting for high-dimensional covariates in sparse precision matrix estimation by $$\ell_1$$-penalization. (English) Zbl 1277.62146
Summary: Motivated by the analysis of genetical genomic data, we consider the problem of estimating a high-dimensional sparse precision matrix adjusting for a possibly large number of covariates, where the covariates can affect the mean value of the random vector. We develop a two-stage estimation procedure to first identify the relevant covariates that affect the means by a joint $$\ell_1$$ penalization. The estimated regression coefficients are then used to estimate the mean values in a multivariate sub-Gaussian model in order to estimate the sparse precision matrix through a $$\ell_1$$-penalized log-determinant Bregman divergence.
Under the multivariate normal assumption, the precision matrix has the interpretation of a conditional Gaussian graphical model. We show that under some regularity conditions, the estimates of the regression coefficients are consistent in the element-wise $$\ell_\infty$$ norm, Frobenius norm and also spectral norm even when $$p \gg n$$ and $$q \gg n$$. We also show that with probability converging to one, the estimate of the precision matrix correctly specifies the zero pattern of the true precision matrix. We illustrate our theoretical results via simulations and demonstrate that the method can lead to improved estimate of the precision matrix. We apply the method to an analysis of a yeast genetical genomic data.

##### MSC:
 62H12 Estimation in multivariate analysis 62P10 Applications of statistics to biology and medical sciences; meta analysis 62Q05 Statistical tables 65C60 Computational problems in statistics (MSC2010)
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