Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators

Consider a class P=P_θ:θ∈Θ of probability measures on a measurable space (X,A), dominated by a σ -finite measure μ. Let f_θ=dP_θ/d_μ, θ\ inΘ, and let θₙ be 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 f_θ: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.