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This paper studies identifiability and convergence behaviors for parametersof multiple types in finite mixtures, and the effects of model fitting withextra mixing components. First, we present a general theory for strongidentifiability, which extends from the previous work of Nguyen [2013] and Chen[1995] to address a broad range of mixture models and to handle matrix-variateparameters. These models are shown to share the same Wasserstein distance basedoptimal rates of convergence for the space of mixing distributions ---$n^{-1/2}$ under $W_1$ for the exact-fitted and $n^{-1/4}$ under $W_2$ for theover-fitted setting, where $n$ is the sample size. This theory, however, is notapplicable to several important model classes, including location-scalemultivariate Gaussian mixtures, shape-scale Gamma mixtures andlocation-scale-shape skew-normal mixtures. The second part of this work isdevoted to demonstrating that for these "weakly identifiable" classes,algebraic structures of the density family play a fundamental role indetermining convergence rates of the model parameters, which display a veryrich spectrum of behaviors. For instance, the optimal rate of parameterestimation in an over-fitted location-covariance Gaussian mixture is preciselydetermined by the order of a solvable system of polynomial equations --- theserates deteriorate rapidly as more extra components are added to the model. Theestablished rates for a variety of settings are illustrated by a simulationstudy.

Identifiability and optimal rates of convergence for parameters of multiple types in finite mixtures