Hello I have a problem with variance estimation for RQMC.
I have $\{x_1,...,x_n\}$ Sobol points. A randomised set $\tilde{\mathfrak{X}}$ is generated by the random variable $\epsilon$. For a randomised set $\tilde{\mathfrak{X}}_i$ we construct a RQMC estimate: \begin{equation} RQMC_{j,n}=\frac{1}{n}\sum_{i=1}^n f(\tilde{\mathfrak{X}}_{i,j}) \end{equation} for $0<j \leqslant r$, where $r$ is the total number of different pseudo-random sequences. Then, $RQMC_{r,n}$ estimation equals: \begin{equation} RQMC_{r,n}=\frac{1}{r}\sum_{j=1}^r RQMC_{j,n}. \end{equation} By independence of the samples we have that: \begin{equation} Var(RQMC_{r,n})=\frac{ Var(RQMC_{j,n})}{r}. \end{equation}
I assume that we have this, because of the unbiasedness: \begin{equation*} Var(RQMC_{r,n})=Var(\frac{1}{r}\sum_{j=1}^rRQMC_{j,n})=\frac{1}{r^2}\sum_{j=1}^rVar(RQMC_{j,n})=\frac{1}{r^2}rVar(RQMC_{j,n})=\frac{1}{r}Var(RQMC_{j,n}). \end{equation*}
However, we have the following variance estimation: \begin{equation*} \widehat{Var}(RQMC_{r,n})=\frac{1}{r(r-1)} \sum_{j=1}^r \Big( RQMC_{j,n}-RQMC_{r,n}\Big)^2\ . \end{equation*}
So I conclude that we have the following unbiased estimator for $RQMC_{j,n}$: \begin{equation*} \widehat{Var}(RQMC_{j,n})=\frac{1}{(r-1)} \sum_{j=1}^r \Big( RQMC_{j,n}-RQMC_{r,n}\Big)^2\ . \end{equation*}
MY PROBLEM IS: that I don't really understand all the steps to obtain $\widehat{Var}(RQMC_{j,n})$ estimation.
It seems that we should use an unbiased estimator of the population variance here: $$ S^2=\frac{1}{r-1}\sum_{j=1}^r(X_j-\bar X)^2 $$
I will really appreciate your help and hints.