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Singularities of a statistical model are the elements of the model'sparameter space which make the corresponding Fisher information matrixdegenerate. These are the points for which estimation techniques such as themaximum likelihood estimator and standard Bayesian procedures do not admit theroot-$n$ parametric rate of convergence. We propose a general framework for theidentification of singularity structures of the parameter space of finitemixtures, and study the impacts of the singularity structures on minimax lowerbounds and rates of convergence for the maximum likelihood estimator over acompact parameter space. Our study makes explicit the deep links between modelsingularities, parameter estimation convergence rates and minimax lower bounds,and the algebraic geometry of the parameter space for mixtures of continuousdistributions. The theory is applied to establish concrete convergence rates ofparameter estimation for finite mixture of skew-normal distributions. This richand increasingly popular mixture model is shown to exhibit a remarkably complexrange of asymptotic behaviors which have not been hitherto reported in theliterature.

Singularity structures and impacts on parameter estimation in finite mixtures of distributions