As most Of distributions in applications have a finite support, we introduce the class of finite distributions with the known shape of their central part and the unknown tails. Furthermore, we use the Huber minimax approach to determine the unknown characteristics of this class. We obtain the least informative distributions minimizing Fisher information for location in the classes of the truncated Gaussian and uniform distributions, and these results give the reasonable values of the thresholds of truncation. The properties of the obtained solutions are discussed.