Here's something I've wondered about for a while, but haven't been able to discover the correct terminology. Say you have a relatively complicated density function that you suspect might have a close approximation as a sum of (properly weighted) simpler density functions. Have such things been studied? I'm particularly interested in reading about any applications.
Here's one example I've found:
You can also use Edgeworth series, if your random variable has a finite mean and variance, which expands the CDF of your random variable in terms of the Gaussian CDF. At first glance it's not quite as tidy conceptually as using a mixture model, but the derivation is quite pretty and it gives you a closed form with very fast decay in the tail terms.
Histogram density estimator is estimating the density with a sum of piecewise functions (density of a uniform).
KDE is using a sum of smooth function (gaussian is an example) (as long as they are positive they can be transformed into a density by normalization)
The use of "mixture" in statistic is about convex combination of densities.
You can do this with mixture modeling. There are a number of R packages on CRAN for doing this. Search for "mixture" at CRAN repository.
