  We formulate multiparameter quantum estimation in the parametric and
semiparametric setting. While the Holevo Cram\'er-Rao bound (CRB) requires no
substantial modifications in moving from the former to the latter, we
generalize the Helstrom CRB appropriately. We show that the Holevo CRB cannot
be greater than twice the generalized Helstrom CRB. We also present a tighter,
intermediate, bound. Finally, we show that for parameters encoded in the first
moments of a Gaussian state there always exists a Gaussian measurement that
gives a classical Fisher information matrix that is one-half of the quantum
Fisher information matrix.
