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In this paper, we aim to estimate block-diagonal covariance matrices forGaussian data in high dimension and in fixed dimension. We first estimate theblock-diagonal structure of the covariance matrix by theoretical and practicalestimators which are consistent. We deduce that the suggested estimator of thecovariance matrix in high dimension converges with the same rate than if thetrue decomposition was known. In fixed dimension , we prove that the suggestedestimator is asymptotically efficient. Then, we focus on the estimation ofsensitivity indices called "Shapley effects", in the high-dimensional Gaussianlinear framework. From the estimated covariance matrix, we obtain an estimatorof the Shapley effects with a relative error which goes to zero at theparametric rate up to a logarithm factor. Using the block-diagonal structure ofthe estimated covariance matrix, this estimator is still available forthousands inputs variables, as long as the maximal block is not too large.

Block-diagonal covariance estimation and application to the Shapley effects in sensitivity analysis