We aim to calculate an expectation $\mu(F)=\Esp{F(\U)}$ for functions $F:[0,1]^d \mapsto R$ using a family of estimators $(\widehat \mu_B)_B$ with a budget of $B$ evaluation points. The standard Monte Carlo method achieves a root mean squared risk of order $1/\sqrt B$, both for a fixed square integrable function $F$ and for the worst-case risk over the class $\mathcal{F}$ of functions with $\|F\|_{L_2}\leq 1$. Using a sequence of Randomized Quasi Monte Carlo (RQMC) methods, in constrast, we have a faster convergence $\sigma_B \ll 1/\sqrt B$ for the risk $\sigma_B=\sigma_B(F)$ when fixing a function $F$, as a difference with the worst-case risk which is still of order $1/\sqrt B$. We address the convergence of quantiles of the absolute error, namely, for a given confidence level $1-\delta$ this is the minimal $\varepsilon$ such that $\P(|\widehat \mu_B(F)-\mu(F)|> \varepsilon)\leq \delta$ holds. We show that a judicious choice of a robust aggregation method coupled with RQMC methods allows reaching improved convergence rates for $\varepsilon$ depending on $\delta$ and $B$, while fixing a function $F$. This study includes a review on concentration bounds for the empirical mean as well as sub-Gaussian mean estimates and is supported by numerical experiments, ranging from bounded $F$ to heavy-tailed $F(\U)$, the latter being well suited to functions $F$ with a singularity. The different methods we have tested are available in a Julia package.}