Pick-freeze Monte Carlo estimators for Sobol indices

Input: a finite product probability law $\mu$, an evaluator $f\in L^2(\mu )$, and a sample size $n\ge 2$. Draw $n$ independent rows. In each row draw six mutually independent vectors $X,X^{\mathrm{\prime }},U,V,W,W^{\mathrm{\prime }}$ with law $\mu$. Reuse $Y=f(X)$; for each $i\in [d]$, evaluate the hybrids $X^{[i]}=(X_i,X_{-i}^{\mathrm{\prime }}),{\tilde{X}}^{[i]}=(X_{-i},X_i^{\mathrm{\prime }}),$ and form $N_i=f(X)f(X^{[i]})-f(U)f(V),T_i=\frac{1}{2}(f(X)-f({\tilde{X}}^{[i]}){)}^2,D=\frac{1}{2}(f(W)-f(W^{\mathrm{\prime }}){)}^2\mathrm{.}$ For the $n$ rows, compute their sample means ${\bar{N}}_i,{\bar{T}}_i,\bar{D}$ and the empirical sample covariance matrix $\widehat{\mathrm{\Sigma }}$ of $Z=(N_1,\dots ,N_d,T_1,\dots ,T_d,D)\mathrm{.}$ Explicitly, for row vectors $Z_r$ and their mean $\bar{Z}$, compute $\widehat{\mathrm{\Sigma }}=\frac{1}{n-1}{\sum }_{r=1}^n(Z_r-\bar{Z}{)}^{\mathsf{T}}(Z_r-\bar{Z})\mathrm{.}$ If $\bar{D}=0$, return status zero-empirical-variance and no normalized index. If $\bar{D}>0$, return the raw ratios ${\widehat{S}}_i={\bar{N}}_i\mathrm{/}\bar{D}$ and ${\widehat{S}}_{T_i}={\bar{T}}_i\mathrm{/}\bar{D}$. Also return the formal first-order plug-in matrix $\widehat{J}\widehat{\mathrm{\Sigma }}{\widehat{J}}^{\mathsf{T}}\mathrm{/}n$, where $\widehat{J}$ is the derivative of the vector ratio map evaluated at $({\bar{N}}_1,\dots ,{\bar{N}}_d,{\bar{T}}_1,\dots ,{\bar{T}}_d,\bar{D})$. No clipping is applied to a negative finite-sample ratio. The $L^2$ input assumption alone does not guarantee a finite population covariance of $Z$ or a valid MCSE interpretation of this matrix.

Pick-freeze covariances recover first and total Sobol numerators makes each numerator and the denominator unbiased for its corresponding variance quantity. The ratios are not asserted unbiased; A pick-freeze ratio can be biased at finite sample size ↗ gives an explicit finite-sample bias witness. Under AC, $f\in L^4(\mu )$ and positive output variance, Pick-freeze Sobol ratios are consistent and have a joint delta-method CLT establishes the joint CLT and the plug-in matrix's asymptotic MCSE interpretation. The empirical matrix formula above is still computed for $L^2$ inputs without these extra assumptions.