How do you test for a mixture of Gaussians distribution?

Question

I have this set of data

Assuming adjustment for its discreteness, this appears to be a mixture of Gaussians distribution (one half-Gaussian). I am able to model mixture of Gaussians, however I want to ensure that the 'cluster' at higher x-values is indeed a statistically significant deviation from the larger half-normal.

Is there a test that I can use to do this?

My first thought was Dixon's Q test - however this requires re-ordering of data, and the 'ordering' of data is already set.

In short, I want to justify the use of a Mixture of Gaussians model more soundly than "it appears to be one."

Answer 1

Comments:

Your link is now broken (at least in my browser). I got the URL, so I will paste it here:

https://i.stack.imgur.com/rgbVh.png

You have a bimodal sample from a discrete distribution. Just from your description, I don't see how you would learn anything trying to a fit mixture of a half-normal and normal.

I tried fitting a Poisson distribution (mean about 0.97, close to 1) to the observed counts for 0 through 7. But that is nowhere near a reasonable fit (according to a chi-squared test). If that had worked, it might have been possible to model the data as a mixture of two Poisson distributions.

Unless you have some knowledge of the process or population that produced these data, leading to a hypothetical model that might be tested, I don't see how to find a useful fit for them.