Minimax bounds for estimation of normal mixtures
Arlene K.H. Kim · Bernoulli · 2014
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.