Given $s_i = E_{\delta \sim p}[ f(x_i + \delta)]$ define $q=\mathrm{Quant}({s_1, \dots, s_n}, \alpha)$ as the $\alpha$ quantile.
One way to obtain a confidence interval for q is to obtain individual confidence intervals for each s_i by sampling $n$ samples for each (e.g. via Hoeffding), combine them with a union bound, and then derive a bound on $q$.
This seems wastefull, since I only want to bound $q$. Is there any way to directly bound $q$? One idea is to ise McDiarmid's inequality but so far this was worse than Hoeffding + union.