Consider finite mixture models of the form $g(x; Q) = \int f(x; \theta) dQ(\theta)$, where f is a parametric density and Q is a discrete probability measure. An important and difficult statistical problem concerns the determination of the number of support points (usually known as components) of Q from a sample of observations from g. For an important class of exponential family models we have the following result: if P has more than p components and Q is an appropriately chosen p-component approximation of P, then $g(x; P) - g(x; Q)$ demonstrates a prescribed sign change behavior, as does the corresponding difference in the distribution functions. These strong structural properties have implications for diagnostic plots for the number of components in a finite mixture.
"Moment-based oscillation properties of mixture models." Ann. Statist. 25 (1) 378 - 386, February 1997. https://doi.org/10.1214/aos/1034276634