  We consider the problem of evaluating dynamic consistency in discrete time
probabilistic filters that approximate stochastic system state densities with
Gaussian mixtures. Dynamic consistency means that the estimated probability
distributions correctly describe the actual uncertainties. As such, the problem
of consistency testing naturally arises in applications with regards to
estimator tuning and validation. However, due to the general complexity of the
density functions involved, straightforward approaches for consistency testing
of mixture-based estimators have remained challenging to define and implement.
This paper derives a new exact result for Gaussian mixture consistency testing
within the framework of normalized deviation squared (NDS) statistics. It is
shown that NDS test statistics for generic multivariate Gaussian mixture models
exactly follow mixtures of generalized chi-square distributions, for which
efficient computational tools are available. The accuracy and utility of the
resulting consistency tests are numerically demonstrated on static and dynamic
mixture estimation examples.
