Stratified Monte Carlo estimator

Let $(\mathsf{X},\mathcal{X},P)$ be a probability space, let $f:\mathsf{X}\to \mathbb{R}$ be measurable with ${\int }_{\mathsf{X}}\mathrm{\mid }f\mathrm{\mid }dP<\mathrm{\infty }$, write $Pf={\int }_{\mathsf{X}}fdP$, and let $H\ge 1$ and $A_1,\dots ,A_H$ be a finite measurable partition of $\mathsf{X}$ with $p_h=P(A_h)>0$. Define the conditional stratum law by $P_h(B)=P(B\cap A_h)\mathrm{/}{p}_h$ for $B\in \mathcal{X}$. Choose fixed integers $n_h\ge 1$ and put $n={\sum }_{h=1}^H{n}_h$. Draw mutually independent variables $X_{hj}$, with $X_{h1},\dots ,X_{h{n}_h}$ IID under $P_h$ as in Identical distribution and IID families, and define ${\bar{f}}_h=n_h^{-1}{\sum }_{j=1}^{n_h}f(X_{hj})$. Expectations and variances use Expectation of a nonnegative or integrable random variable and Moments, variance, and covariance on a probability space. The stratified Monte Carlo estimator of $Pf$ is

${\widehat{\mu }}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}}={\sum }_{h=1}^H{p}_h{\bar{f}}_h\mathrm{.}$

The allocation $(n_1,\dots ,n_H)$ is part of the design. A variance comparison must specify it and the comparator.