  This paper deals with minimax rates of convergence for estimation of density
functions on the real line. The densities are assumed to be location mixtures
of normals, a global regularity requirement that creates subtle difficulties
for the application of standard minimax lower bound methods. Using novel
Fourier and Hermite polynomial techniques, we determine the minimax optimal
rate - slightly larger than the parametric rate - under squared error loss. For
Hellinger loss, we provide a minimax lower bound using ideas modified from the
squared error loss case.
