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!!

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    $\begingroup$ This question is broader than may seem to you, because your formulation comprises all questions about kernel smooths--$Z$ is the kernel--and even all questions about sums of random variables. Please edit it to explain the focus of your interest. $\endgroup$
    – whuber
    Mar 12 '14 at 16:27
  • $\begingroup$ OK, thanks, I will re-edit it and try to make it more specialized. $\endgroup$
    – Jingjings
    Mar 12 '14 at 17:37
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    $\begingroup$ Are variables produced by multiplying random variables actually mixtures? This seems at odds with the way mixtures are usually defined. For example, the product of two Gaussian random variables is distributed as a linear combination of two Chi-square random variables, rather than a mixture of Gaussians. $\endgroup$
    – dimitriy
    Mar 12 '14 at 17:59
  • $\begingroup$ Other special cases here. $\endgroup$
    – dimitriy
    Mar 12 '14 at 18:09
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    $\begingroup$ Please don't ask completely new questions in comments - either modify your question or ask a new one. $\endgroup$
    – Glen_b
    Mar 13 '14 at 9:18

In statristics, 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:

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.


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