Efficient estimation of Sobol indices

Chatterjee's method provides arguably the simplest way to estimate first order Sobol indices, as the empirical auto-correlation of the response variable obtained upon re-ordering the data by increasing values of the inputs. This idea can be extended to higher lags of auto-correlation, thus providing several competing estimators of the same parameter. We show that these estimators can be combined in a simple manner to achieve the theoretical variance efficiency bound asymptotically.