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

## TL;DR

This paper develops consistent estimators for block-diagonal covariance matrices in high-dimensional Gaussian data and applies them to efficiently estimate Shapley effects in sensitivity analysis, even with thousands of variables.

## Contribution

It introduces new estimators for block-diagonal covariance matrices that are both consistent and efficient, enabling scalable sensitivity analysis in high dimensions.

## Key findings

- Estimator converges at the same rate as if the true structure was known.
- Estimator is asymptotically efficient in fixed dimension.
- Allows estimation of Shapley effects for thousands of variables.

## Abstract

In this paper, we aim to estimate block-diagonal covariance matrices for Gaussian data in high dimension and in fixed dimension. We first estimate the block-diagonal structure of the covariance matrix by theoretical and practical estimators which are consistent. We deduce that the suggested estimator of the covariance matrix in high dimension converges with the same rate than if the true decomposition was known. In fixed dimension , we prove that the suggested estimator is asymptotically efficient. Then, we focus on the estimation of sensitivity indices called "Shapley effects", in the high-dimensional Gaussian linear framework. From the estimated covariance matrix, we obtain an estimator of the Shapley effects with a relative error which goes to zero at the parametric rate up to a logarithm factor. Using the block-diagonal structure of the estimated covariance matrix, this estimator is still available for thousands inputs variables, as long as the maximal block is not too large.