Nonparametric estimation of the first order Sobol indices with bootstrap bandwidth
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Suppose that Y = m(X_1, ..., X_p), where (X_1, ..., X_p) are inputs, Y is an output, and m is an unknown function. The Sobol indices gauge sensitivity each X against Y by estimating the regression curve's variability between them. In this paper, we estimate it with a kernel-based methodology. Under some regularity conditions, the mean squared error for the first order Sobol indices have an asymptotically parametric behavior, with bandwidth equivalent to n^-1/4 where n is the sample size. For finite samples, the cross-validation method produces a structural bias. To correct this, we propose a bootstrap procedure which reconstruct the model residuals and re-estimate the nonparametric regression curve. With the new curve, the procedure corrects the bias in the Sobol index. We present simulated two numerical examples with complex functions to assess the performance of our procedure.
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