  Quantum estimation theory provides optimal observations for various
estimation problems for unknown parameters in the state of the system under
investigation. However, the theory has been developed under the assumption that
every observable is available for experimenters. Here, we generalize the theory
to problems in which the experimenter can use only locally accessible
observables. For such problems, we establish a Cram{\'e}r-Rao type inequality
by obtaining an explicit form of the Fisher information as a reciprocal lower
bound for the mean square errors of estimations by locally accessible
observables. Furthermore, we explore various local quantum estimation problems
for composite systems, where non-trivial combinatorics is needed for obtaining
the Fisher information.
