This paper studies the estimation of a large covariance matrix. We introducea novel procedure called ChoSelect based on the Cholesky factor of the inversecovariance. This method uses a dimension reduction strategy by selecting thepattern of zero of the Cholesky factor. Alternatively, ChoSelect can beinterpreted as a graph estimation procedure for directed Gaussian graphicalmodels. Our approach is particularly relevant when the variables under studyhave a natural ordering (e.g. time series) or more generally when the Choleskyfactor is approximately sparse. ChoSelect achieves non-asymptotic oracleinequalities with respect to the Kullback-Leibler entropy. Moreover, itsatisfies various adaptive properties from a minimax point of view. We alsointroduce and study a two-stage procedure that combines ChoSelect with theLasso. This last method enables the practitioner to choose his own trade-offbetween statistical efficiency and computational complexity. Moreover, it isconsistent under weaker assumptions than the Lasso. The practical performancesof the different procedures are assessed on numerical examples.
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