Journal of the
Korean Mathematical Society JKMS

The least informative distributions minimizing Fisher information for location are obtained in the classes of continuously differentiable and piece-wise continuously differentiable densities with the additional restrictions on their values at the median and mode of population in the point and interval forms. The structure of these optimal solutions depends both on the assumptions of smoothness and form of characterizing restrictions of the class of distributions: in the class of continuously differentiable densities, the least informative distributions are finite and have the {\it cosine}-type form, and, in the class of piece-wise continuously differentiable densities, the least informative densities have {\it exponential}-type tails, the Laplace density in particular. The dependence of optimal solutions on the assumptions of symmetry is also analyzed.

Keywords: minimax approach, robustness, least informative distributions, location parameter

MSC numbers: Primary 62F35, secondary 62B10