GENERALIZED MAXIMUM LIKELIHOOD ESTIMATION OF NORMAL MIXTURE DENSITIES
Cun‐Hui Zhang · Statistica Sinica · 2009
We study the generalized maximum likelihood estimator of location and location-scale mixtures of normal densities. A large deviation inequality is ob- tained which provides the convergence rate n ip/(2+2p) (logn) κ 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 i1/2 (logn) κ 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 mix- tures. 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.