Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators

Consider a class P=Pθ:θ∈Θ of probability measures on a measurable space (X,A), dominated by a σ -finite measure μ. Let fθ=dPθ/dμ, θ inΘ, and let θnbe a maximum likelihood estimator based on n independent observations from Pθ0 , θ0∈Θ. We use results from empirical process theory to obtain convergence for the Hellinger distance h(fθ̂n , fθ0 ), under certain entropy conditions on the class of densities fθ:θ∈Θ The examples we present are a model with interval censored observations, smooth densities, monotone densities and convolution models. In most examples, the convexity of the class of densities is of special importance.