On Inequalities Relating the Characteristic Function and Fisher Information

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.