Honest confidence regions and optimality in high-dimensional precision matrix estimation

Jana Janková () and Sara Geer ()
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Jana Janková: Seminar for Statistics, ETH Zürich
Sara Geer: Seminar for Statistics, ETH Zürich

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2017, vol. 26, issue 1, No 7, 143-162

Abstract: Abstract We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters p can be much larger than the sample size. We show that the novel estimator achieves minimax rates in supremum norm and the low-dimensional components of the estimator have a Gaussian limiting distribution. These results hold uniformly over the class of precision matrices with row sparsity of small order $$\sqrt{n}/\log p$$ n / log p and spectrum uniformly bounded, under a sub-Gaussian tail assumption on the margins of the true underlying distribution. Consequently, our results lead to uniformly valid confidence regions for low-dimensional parameters of the precision matrix. Thresholding the estimator leads to variable selection without imposing irrepresentability conditions. The performance of the method is demonstrated in a simulation study and on real data.

Keywords: Precision matrix; Sparsity; Inference; Asymptotic normality; Confidence regions; 62J07; 62F12 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s11749-016-0503-5

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