 Solution

For independent draws $X_i$ from a normalized proposal $g$, use the importance sampling identityThus the unbiased estimator and its variance areThe probability support condition is $g>0$ wherever $\theta f\ne 0$, and the variance formula requires a finite second moment. Since both probability densities are normalized, no self-normalization of weights is needed.

In the proposed example, normalize $g$ to $g(x)=3x^2$ on $(0,1)$. It is sampled by $X=U^{1/3}$ for uniform $U$. Taking $\theta (x)=x$ givesAs a check, ${\mu }_g=\frac{4}{\pi }{\int }_0^{1}x\sqrt{1-x^2}dx=4/(3\pi )$. The second moment of one weighted observation isso $\mathrm{Var}({\hat{\mu }}_g)=16/(9{\pi }^{2}n)$. The weight's singularity at zero does not make this variance infinite.

To prove the optimal proposal rule, apply the Cauchy-Schwarz inequality:Equality holds when the two factors are proportional almost everywhere, giving the minimum-variance importance distributionFor the present nonnegative integrand, normalization givesUnder this proposal, $xf(x)/g_0(x)=4/(3\pi )$ is constant, so the variance is exactly zero. If sampling is wanted, its distribution function is $1-(1-x^2{)}^{3/2}$, with inverse $X=\sqrt{1-(1-U{)}^{2/3}}$. In general the optimal proposal's normalizing constant can itself be the unknown integral, limiting this zero-variance construction's practical use.