  The large-sample properties of likelihood-based statistical inference under
mixture models have received much attention from statisticians. Although the
consistency of the nonparametric MLE is regarded as a standard conclusion, many
researchers ignore the precise conditions required on the mixture model. An
incorrect claim of consistency can lead to false conclusions even if the
mixture model under investigation seems well behaved. Under a finite normal
mixture model, for instance, the consistency of the plain MLE is often
erroneously assumed in spite of recent research breakthroughs. This paper
streamlines the consistency results for the nonparametric MLE in general, and
in particular for the penalized MLE under finite normal mixture models.
