Independent splitting estimator moments and work

Under ${\pi }_{\mathrm{\ell }-1}$, the indicator ${\mathbf{1}}_{A_{\mathrm{\ell }}}$ is Bernoulli with success probability $P(A_{\mathrm{\ell }})\mathrm{/}P(A_{\mathrm{\ell }-1})=p_{\mathrm{\ell }}$. Thus [F4] gives $\mathbb{E}{\widehat{p}}_{\mathrm{\ell }}=p_{\mathrm{\ell }}$ and $\mathbb{E}{\widehat{p}}_{\mathrm{\ell }}^2=p_{\mathrm{\ell }}^2+p_{\mathrm{\ell }}(1-p_{\mathrm{\ell }})\mathrm{/}N$ at each level.

The level arrays are independent by [F3], so $\mathbb{E}\widehat{p}={\prod }_{\mathrm{\ell }}{p}_{\mathrm{\ell }}=p$ by [F2], and $\mathbb{E}{\widehat{p}}^2={\prod }_{\mathrm{\ell }}[p_{\mathrm{\ell }}^2+p_{\mathrm{\ell }}(1-p_{\mathrm{\ell }})\mathrm{/}N]$. Subtracting $p^2={\prod }_{\mathrm{\ell }}{p}_{\mathrm{\ell }}^2$ gives the displayed variance and, after division by $p^2>0$, the relative-variance formula. The work equality is the $LN$ declared draws weighted by their level costs in [F3].

Put $a_{\mathrm{\ell }}=p_{\mathrm{\ell }}^2$ and $v_{\mathrm{\ell }}=p_{\mathrm{\ell }}(1-p_{\mathrm{\ell }})\mathrm{/}N$. Then $0\le a_{\mathrm{\ell }}\le 1$ and $0\le v_{\mathrm{\ell }}\le 1\mathrm{/}(4N)$. By step 2.1 and finite product expansion, $\mathrm{Var}(\widehat{p})={\prod }_{\mathrm{\ell }=1}^L(a_{\mathrm{\ell }}+v_{\mathrm{\ell }})-{\prod }_{\mathrm{\ell }=1}^L{a}_{\mathrm{\ell }}={\sum }_{\mathrm{\varnothing }\mathrm{\ne }S\subseteq \{1,\dots ,L\}}\left({\prod }_{\mathrm{\ell }\in S}{v}_{\mathrm{\ell }}\right)\left({\prod }_{j\mathrm{\notin }S}{a}_j\right)\le {\sum }_{\mathrm{\varnothing }\mathrm{\ne }S\subseteq \{1,\dots ,L\}}{\left(\frac{1}{4N}\right)}^{\mathrm{\mid }S\mathrm{\mid }}={(1+\frac{1}{4N})}^L-1.$

For fixed $L$, expansion of the finite binomial gives $(1+1\mathrm{/}(4N){)}^L-1\le \frac{L}{4N}(1+1\mathrm{/}(4N){)}^{L-1}=O(L\mathrm{/}N)$. The relative formula in step 2.1 still contains $(1-p_{\mathrm{\ell }})\mathrm{/}(N{p}_{\mathrm{\ell }})$, so this absolute bound does not give a rarity-uniform relative bound.