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Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators

Sara van de Geer
The Annals of Statistics
Vol. 21, No. 1 (Mar., 1993), pp. 14-44
Stable URL: http://www.jstor.org/stable/3035578
Page Count: 31
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Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators
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Abstract

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 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 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.

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