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We study the numerical integration of functions from isotropic Sobolev spaces$W_p^s([0,1]^d)$ using finitely many function evaluations within randomizedalgorithms, aiming for the smallest possible probabilistic error guarantee$\varepsilon > 0$ at confidence level $1-\delta \in (0,1)$. For spacesconsisting of continuous functions, non-linear Monte Carlo methods with optimalconfidence properties have already been known, in few cases even linear methodsthat succeed in that respect. In this paper we promote a new method calledstratified control variates (SCV) and by it show that already linear methodsachieve optimal probabilistic error rates in the high smoothness regime withoutthe need to adjust algorithmic parameters to the uncertainty $\delta$. We alsoanalyse a version of SCV in the low smoothness regime where $W_p^s([0,1]^d)$may contain functions with singularities. Here, we observe a polynomialdependence of the error on $\delta^{-1}$ which cannot be avoided for linearmethods. This is worse than what is known to be possible using non-linearalgorithms where only a logarithmic dependence on $\delta^{-1}$ occurs if wetune in for a specific value of $\delta$.

Linear Monte Carlo quadrature with optimal confidence intervals