  A relationship between the Fisher information and the characteristic function
is established with the help of two inequalities. A necessary and sufficient
condition for equality is found. These results are used to determine the
asymptotic efficiency of a distributed estimation algorithm that uses constant
modulus transmissions over Gaussian multiple access channels. The loss in
efficiency of the distributed estimation scheme relative to the centralized
approach is quantified for different sensing noise distributions. It is shown
that the distributed estimator does not incur an efficiency loss if and only if
the sensing noise distribution is Gaussian.
