Minimax Bounds for Estimation of Normal Mixtures

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.