= Solution
By invariance of maximum likelihood,
$$
\widehat h=10^{\widehat\theta/5}
=\exp(\widehat\theta/\alpha).
$$
Because $\widehat\theta-\theta\sim N(0,\sigma_\theta^2)$, the ratio $\widehat h/h$ is <log-normal distribution>[log-normal]. Its exact variance is
$$
\operatorname{Var}\!\left(\frac{\widehat h}{h}\right)
=e^{\sigma_\theta^2/\alpha^2}
\left(e^{\sigma_\theta^2/\alpha^2}-1\right)
=\frac{\sigma_\theta^2}{\alpha^2}+O(\sigma_\theta^4).
$$
As $N_{\mathrm{SN}},G,N_{\mathrm{LMC}}\to\infty$, all sampling terms vanish but $V_1$ retains $\sigma_{\mathrm{LMC}}^2$. The dominant remaining uncertainty is therefore the parallax calibration of the LMC distance modulus.
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