The Shapley effects are global sensitivity indices: they quantify the impact of each input variable on the output variable in a model. In this work, we suggest new estimators of these sensitivity indices. When the input distribution is known, we investigate the already existing estimator defined in [SNS16] and suggest a new one with a lower variance. Then, when the distribution of the inputs is unknown, we extend these estimators. We provide asymptotic properties of the estimators studied in this article. We also apply one of these estimators to a real data set.

In this paper, we study sensitivity indices for independent groups of variables and we look at the particular case of block-additive models. We show in this case that most of the Sobol indices are equal to zero and that Shapley effects can be estimated more efficiently. We then apply this study to Gaussian linear models, and we provide an efficient algorithm to compute the theoretical sensitivity indices. In numerical experiments, we show that this algorithm compares favourably to other existing methods. We also use the theoretical results to improve the estimation of the Shapley effects for general models, when the inputs form independent groups of variables.

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