  In a previous paper [M. Tsang, Phys. Rev. A 99, 012305 (2019)], I proposed a
quantum limit to the estimation of object moments in subdiffraction incoherent
optical imaging. In this sequel, I prove the quantum limit rigorously by
infinite-dimensional analysis. A key to the proof is the choice of an
unfavorable parametric submodel to give a bound for the semiparametric problem.
By generalizing the quantum limit for a larger class of moments, I also prove
that the measurement method of spatial-mode demultiplexing (SPADE) with just
one or two modes is able to achieve the quantum limit. For comparison, I derive
a classical bound for direct imaging using the parametric-submodel approach,
which suggests that direct imaging is substantially inferior.
