We propose two novel unbiased estimators of the integral $\int_{[0,1]^{s}}f(u) du$ for a function $f$, which depend on a smoothness parameter $r\in\mathbb{N}$. The first estimator integrates exactly the polynomials of degrees $p<r$ and achieves the optimal error $n^{-1/2-r/s}$ (where $n$ is the number of evaluations of $f$) when $f$ is $r$ times continuously differentiable. The second estimator is computationally cheaper but it is restricted to functions that vanish on the boundary of $[0,1]^s$. The construction of the two estimators relies on a combination of cubic stratification and control ariates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of $n$.

翻译：我们针对函数$f$的积分$\int_{[0,1]^{s}}f(u) du$提出两种新型无偏估计量，其构造依赖于光滑性参数$r\in\mathbb{N}$。第一种估计量可精确积分次数$p<r$的多项式，且当$f$具有$r$阶连续可微性时达到最优误差$n^{-1/2-r/s}$（其中$n$为对$f$的求值次数）。第二种估计量计算成本较低，但仅适用于在$[0,1]^s$边界上取值为零的函数。两种估计量的构造均基于三次分层方法与数值导数控制变量的结合。数值实验表明，即使对于中等大小的$n$值，这两种方法仍展现出良好的性能。