  The theory of semiparametric estimation offers an elegant way of computing
the Cram\'er-Rao bound for a parameter of interest in the midst of infinitely
many nuisance parameters. Here I apply the theory to the problem of moment
estimation for incoherent imaging under the effects of diffraction and photon
shot noise. Using a Hilbert-space formalism designed for Poisson processes, I
derive exact semiparametric Cram\'er-Rao bounds and efficient estimators for
both direct imaging and a quantum-inspired measurement method called
spatial-mode demultiplexing (SPADE). The results establish the superiority of
SPADE even when little prior information about the object is available.
