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This is a terminology question motivated by a review that I got on a paper. In the following I believe that $y$ would be considered to be distributed according to a mixture distribution:

$$y \sim \left\{ \begin{array}{ll} f_1(\theta_1) & \mbox{with } p ~ \mbox{probability} \\ f_2(\theta_2) & \mbox{with } (1 - p) ~ \mbox{probability} \end{array} \right. $$

where $f_1$ and $f_2$ are probability distributions. But would it be correct to say that $x$ is also distributed according to a mixture distribution?

$$ x \sim \text{min}(z_1,z_2) \\ z_1 \sim f_1(\theta_1) \\ z_2 \sim f_2(\theta_2)$$

If not, is there a better term? Or, at least, a general term for a distribution that is some combination of two other distributions?

(I mean, $x$ is certainly the result of "mixing together" $f_1$ and $f_2$ in some way, but I don't know if this is enough to call it a mixture... :)

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    $\begingroup$ I believe there is no general term for "some combination" of distributions simply because the concept is far too broad to be useful. $\endgroup$ – whuber Sep 10 '14 at 15:27
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    $\begingroup$ I would say most latent variable models qualify for your "mixing" behaviour, but this does not help much because about everything is a latent variable model! $\endgroup$ – Xi'an Nov 25 '14 at 15:29
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I don't think that counts as a mixture, no.

A mixture distribution draws from among a collection of distributions according to a random variable from another distribution (as in your first example, though of course they can be more complex than Bernoulli mixtures, including continuous mixtures).

The $\min$ case doesn't seem to have the characteristics I'd normally understand as being a mixture.

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  • $\begingroup$ So do you have a suggestion for what to call it? A compound distribution? A hybrid distribution? A blend distribution? :) $\endgroup$ – Rasmus Bååth Sep 10 '14 at 19:06
  • $\begingroup$ No, I don't, sorry. If I had been able to think of a good specific term, I'd mention it. If I find one I'll let you know. $\endgroup$ – Glen_b -Reinstate Monica Sep 10 '14 at 21:16
  • $\begingroup$ The best I could think of was "the distribution of the minimum" ... which is no help. $\endgroup$ – Glen_b -Reinstate Monica Oct 2 '14 at 10:54

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