  Recent research has established sufficient conditions for finite mixture
models to be identifiable from grouped observations. These conditions allow the
mixture components to be nonparametric and have substantial (or even total)
overlap. This work proposes an algorithm that consistently estimates any
identifiable mixture model from grouped observations. Our analysis leverages an
oracle inequality for weighted kernel density estimators of the distribution on
groups, together with a general result showing that consistent estimation of
the distribution on groups implies consistent estimation of mixture components.
A practical implementation is provided for paired observations, and the
approach is shown to outperform existing methods, especially when mixture
components overlap significantly.
