  This paper studies identifiability and convergence behaviors for parameters
of multiple types in finite mixtures, and the effects of model fitting with
extra mixing components. First, we present a general theory for strong
identifiability, which extends from the previous work of Nguyen [2013] and Chen
[1995] to address a broad range of mixture models and to handle matrix-variate
parameters. These models are shown to share the same Wasserstein distance based
optimal 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 the
over-fitted setting, where $n$ is the sample size. This theory, however, is not
applicable to several important model classes, including location-scale
multivariate Gaussian mixtures, shape-scale Gamma mixtures and
location-scale-shape skew-normal mixtures. The second part of this work is
devoted to demonstrating that for these "weakly identifiable" classes,
algebraic structures of the density family play a fundamental role in
determining convergence rates of the model parameters, which display a very
rich spectrum of behaviors. For instance, the optimal rate of parameter
estimation in an over-fitted location-covariance Gaussian mixture is precisely
determined by the order of a solvable system of polynomial equations --- these
rates deteriorate rapidly as more extra components are added to the model. The
established rates for a variety of settings are illustrated by a simulation
study.
