How to calculate Quasi-Monte Carlo integration error when sampling with Sobol's sequence? My understanding is that QMC integration using random sampling will converge with $O(\frac{1}{\sqrt{n}})$, while using Sobol's sampling will converge with $O(\frac{(\log{n})^d}{n})$. However I'm having trouble converting this big O notation into concrete error bounds.
This source gives an equation for calculating error bounds when random sampling. Given a set of random samples with variance $\sigma^2$ drawn from volume $V$ and a chosen confidence level $\gamma \in (0,1)$, we can find the bounding integration error from $\epsilon_\gamma = V\sigma\Phi^{-1}(\gamma)\frac{1}{\sqrt{n}}$, which follows from the central limit theorem.
Is finding the error bounds when Sobol sampling just as easy? Should I be using this? $\epsilon_\gamma = V\sigma\Phi^{-1}(\gamma)\frac{(\log{n})^d}{n}$
In my empirical tests of 1-dimensional sequences it seems to be a reasonable (if anything, too high) upper bound, but I want to make sure I'm theoretically grounded here since higher dimensions are harder to test.


You might find the paper Quasi-random sequences and their discrepancies by Morokoff and Caflisch (1994) useful.
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.298.5482&rep=rep1&type=pdf
They give bounds of the form
$$
\left|\int_{[0,1]^d}f(x)dx - \frac{1}{N}\sum_{i=1}^N f(x_i) \right| \le V(f) D_N^* \, ,
$$
where $V(f)$ is the variation of $f$ in the sense of Hardy and Krause and $D_N^*$ is the star-discrepancy with
$$
D_N^* = \mathcal{O}\left( \frac{(\log N)^d}{N} \right) \, .
$$
The paper also provides some guidance on the missing pre-factor.  In general these bounds are only useful if the number of points is very large.  Moreover, the variation $V(f)$ may be quite complicated and intractable to compute.
I don't know how well this would work, but a simple solution that might be worth trying is resampling your points to compute bootstrap-type estimates of the confidence intervals.
