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Summary of the paper This paper presents a semi-parametric tuning-free procedure for estimating sparse concentration matrices. This method is applicable to the elliptical distribution family, while most of its competitors only apply to the sub-Gaussian distribution family. The procedure, called ALICE, learns the precision matrix column by column in a similar fashion than the CLIME (Cai et al, 2011), yet with important modifications: a first step is designed to learn the correlation matrix and the associated variances/standard deviations by means of the Kendall's Tau statistic as proposed in Liu et al, 2012. Then, the standard deviations are estimated through a recent proposal of Catoni (2012). In the second step, the inverse correlation is recovered by plugin-in the correlation estimated in the first step in a convex program similar to the CLIME, yet with a modification that allows for some calibration between the columns.