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A constrained $$\ell _{1}$$ minimization approach to sparse precision matrix estimation. (English) Zbl 1232.62087
Summary: This article proposes a constrained $$\ell _{1}$$ minimization method for estimating a sparse inverse covariance matrix based on a sample of $$n$$ iid $$p$$-variate random variables. The resulting estimator is shown to have a number of desirable properties. In particular, the rate of convergence between the estimator and the true $$s$$-sparse precision matrix under the spectral norm is $$s \sqrt {\log p/n}$$ when the population distribution has either exponential-type tails or polynomial-type tails. We present convergence rates under the elementwise $$\ell _{\infty }$$ norm and Frobenius norm. In addition, we consider graphical model selection. The procedure is easily implemented by linear programming. Numerical performance of the estimator is investigated using both simulated and real data. In particular, the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods.

##### MSC:
 62H12 Estimation in multivariate analysis 62F12 Asymptotic properties of parametric estimators 90C05 Linear programming 62P10 Applications of statistics to biology and medical sciences; meta analysis 65C60 Computational problems in statistics (MSC2010)
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