  We study the geometric measure of entanglement (GM) of pure symmetric states
related to rank-one positive-operator-valued measures (POVMs) and establish a
general connection with quantum state estimation theory, especially the maximum
likelihood principle. Based on this connection, we provide a method for
computing the GM of these states and demonstrate its additivity property under
certain conditions. In particular, we prove the additivity of the GM of pure
symmetric multiqubit states whose Majorana points under Majorana representation
are distributed within a half sphere, including all pure symmetric three-qubit
states. We then introduce a family of symmetric states that are generated from
mutually unbiased bases (MUBs), and derive an analytical formula for their GM.
These states include Dicke states as special cases, which have already been
realized in experiments. We also derive the GM of symmetric states generated
from symmetric informationally complete POVMs (SIC~POVMs) and use it to
characterize all inequivalent SIC~POVMs in three-dimensional Hilbert space that
are covariant with respect to the Heisenberg--Weyl group. Finally, we describe
an experimental scheme for creating the symmetric multiqubit states studied in
this article and a possible scheme for measuring the permanent of the related
Gram matrix.
