Nonparametric estimation of the first order Sobol indices with bootstrap bandwidth

Suppose that $Y = \psi(X_1, \ldots, X_p)$, where $(X_1,\ldots, X_p)^\top$ are random inputs, $Y$ is the output, and $\psi(\cdot)$ is an unknown link function. The Sobol indices gauge the sensitivity of each $X$ against $Y$ by estimating the regression curve's variability between them. In this paper, we estimate these curves with a kernel-based method. The method allows to estimate the first order indices when the link between the independent and dependent variables is unknown. The kernel-based methods need a bandwidth to average the observations. For finite samples, the cross-validation method is famous to decide this bandwidth. However, it produces a structural bias. To remedy this, we propose a bootstrap procedure which reconstruct the model residuals and re-estimate the non-parametric regression curve. With the new set of curves, the procedure corrects the bias in the Sobol index. To test the developed method, we implemented simulated numerical examples with complex functions.