Is there other types of mixture distribution besides the normal mixture

Question

There are quite a lot of study on the normal mixture distributions, say, $X=Y*Z$,where $Z$ is a normal r.v. and Y is a r.v. follows other distributions and $Y$ and $Z$ are independent. Some well-known distributions are Normal inverse Gaussian, generalized Hyperbolic etc.

I would like to know that in probability, is there some other mixture distributions in such form (product of two independent r.v.s) that are well-studied? For example, let $Z$ is possion or exponential and then is there some well-studied probablity distribution can be generated from this form.

If possible, could anyone so kind give me some textbook or monograph on this topic? Thanks so much!!

Answer 1

In statistics, a mixture of distributions is typically defined as a density written as a finite or countably finite weighted sum of other densities $$f(x)=\sum_{i=1}^k \omega_i f_i(x)\qquad\sum_{i=1}^k \omega_i=1\quad 0<\omega_i$$Those component densities $f_i$ are usually from standard families, e.g., Gaussians, but in essence they can be essentially any density. However, Gaussians are favoured, if only because of a result of Chu (1973) that shows that any even, unimodal, continuous density is a scale mixture of Gaussians. (Except that $h$ is not necessarily a density.) See, e.g., West (1987) for details.

References on mixture estimation are numerous, among which I can recommend:

• Frühwirth-Schnatter, S. (2006), Finite Mixture and Markov Switching Models, Springer
• McLachlan, G. and Peel, D. (2000) Finite Mixture Models, Wiley

Scale mixtures are a misnomer in that they characterise distributions with a scale factor that is integrated out wrt a given distribution, for instance a Gaussian. Meaning the density is of the form [in dimension one] $$f(x) = \int_0^\infty \tau^{-1}\varphi(x/\tau) h(\tau)\text{d}\tau$$ where $\varphi$ and $h$ can be any density [under integrability restrictions]. Except that Gaussians are special in that any symmetric (even), unimodal (at zero), continuous density is a scale mixture of Gaussians as in the above [but with some cases when $h$ is not a density]. See e.g. West (1987) for details.