Existence and consistency of maximum likelihood in upgraded mixture models. (English) Zbl 0752.62026

Summary: Suppose one observes a sample of size \(m\) from the mixture density \(\int p(x\mid z)d\eta(z)\) and a sample of size \(n\) from the distribution \(\eta\). The kernel \(p(x\mid z)\) is known. We show existence of the maximum likelihood estimator for \(\eta\), characterize its support, and prove consistency as \(m\), \(n\to\infty\).


62F12 Asymptotic properties of parametric estimators
62G20 Asymptotic properties of nonparametric inference
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