GENERALIZED MAXIMUM LIKELIHOOD ESTIMATION OF NORMAL MIXTURE DENSITIES

We study the generalized maximum likelihood estimator of location and location-scale mixtures of normal densities. A large deviation inequality is obtained which provides the convergence rate n(-p/(2+2p)) (log n)(kappa p) in the Hellinger distance for mixture densities when the mixing distributions have bounded finite p-th weak moment, p > 0, and the convergence rate n(-1/2) (log n)(kappa) when the mixing distributions have an exponential tail uniformly. Our results are applicable to the estimation of the true density of independent identically distributed observations from a normal mixture, as well as the estimation of the average marginal densities of independent not identically distributed observations from different normal mixtures. The validity of our results for mixing distributions with p-th weak moment, 0 < p < 2, and for not identically distributed observations, is of special interest in compound estimation and other problems involving sparse normal means.