  Measurement estimation bounds for local quantum multiparameter estimation,
which provide lower bounds on possible measurement uncertainties, have so far
been formulated in two ways: by extending the classical Cram\'er--Rao bound
(e.g., the quantum Cram\'er--Rao bound and the Nagaoka Cram'er--Rao bound) and
by incorporating the parameter estimation framework with the uncertainty
principle, as in the Lu--Wang uncertainty relation. In this work, we present a
general framework that allows a direct comparison between these different types
of estimation limits. Specifically, we compare the attainability of the Nagaoka
Cram\'er--Rao bound and the Lu--Wang uncertainty relation, using analytical and
numerical techniques. We show that these two limits can provide different
information about the physically attainable precision. We present an example
where both limits provide the same attainable precision and an example where
the Lu--Wang uncertainty relation is not attainable even for pure states. We
further demonstrate that the unattainability in the latter case arises because
the figure of merit underpinning the Lu--Wang uncertainty relation (the
difference between the quantum and classical Fisher information matrices) does
not necessarily agree with the conventionally used figure of merit (mean
squared error). The results offer insights into the general attainability and
applicability of the Lu--Wang uncertainty relation. Furthermore, our proposed
framework for comparing bounds of different types may prove useful in other
settings.
