On minimum Fisher information distributions with restricted support and fixed variance

Fisher information is of key importance in estimation theory. It also serves in inference problems as well as in the interpretation of many physical processes. The mean-squared estimation error for the location parameter of a distribution is bounded by the inverse of the Fisher information associated with this distribution. In this paper we look for minimum Fisher information distributions with a restricted support. More precisely, we study the problem of minimizing the Fisher information in the set of distributions with fixed variance defined on a bounded subset image of image or on the positive real line. We show that the solutions of the underlying differential equation can be expressed in terms of Whittaker functions. Then, in the two considered cases, we derive the explicit expressions of the solutions and investigate their behavior. We also characterize the behavior of the minimum Fisher information as a function of the imposed variance.