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Through simulation I'm comparing several methods for isolating a signal distribution from complicated, noisy background (two components in a mixture). What I need is a good PDF model for this noisy background and for the signal (for simplicity I'd like the signal and background PDFs to be described by the same model, just with more parameters for the background; I'm willing to change this). I'm currently using a Fourier-like family of PDFs on the interval $x \in [0,\pi]$: $$ pdf(x) \propto 1 + \sum_{k=1}^{K} \frac{1}{k} \left( |a_k| + a_k \cos(k x) \right) $$ This has several nice properties:

  1. It has an analytic CDF, which makes it easy to sample from the distribution
  2. For small $K$, this can be used as a smooth signal shape
  3. For large $K$ and randomly generated $a_k$, e.g. from an exponential distribution, the shape is chaotic (in a non-technical sense) and good for a background shape:

enter image description here

However, it has one drawback: since it is not a member of the exponential family, it is costly to determine its parameters via likelihood maximization (which I need to do not for the noisy background, but for the signal with small $K$ and only a few parameters).

Is there a similar exponential-family model that I can use for signal and for noisy background?. Alternatively, I could keep this model for the background and switch my signal model -- is there an exponential-family model with one or more parameters on a bounded interval, whose parameters can be determined quickly with likelihood maximization?

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  • $\begingroup$ From some research, it seems like if I switched to the interval [-pi,pi], I could used the wrapped normal distribution for my signal. I'm not quite sure if that's in the exponential family (likely not), but efficient parameter estimation looks possible. $\endgroup$
    – jwimberley
    Jan 9, 2017 at 16:48
  • $\begingroup$ Actually, I am having good success with using the von Mises distribution for my simple signal shape, and the above Fourier series pdf for my noisy background shape. $\endgroup$
    – jwimberley
    Jan 10, 2017 at 2:42

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I'll answer my question because I found a good option. The von Mises distribution on the bounded interval $[0,2\pi]$ is a member of the exponential family whose parameters can be estimated using standard trig functions and a quick numerical solution to an equation involving Bessel's functions. The CDF is not analytic, but that's OK because several R packages, such as CircStats, implement rvm, dvm, pvm, etc., so I didn't need to manually implement the random number generator. My "noisy" background was easy to adapt to the interval $[0,2\pi]$. I don't get to use the same model for signal and background, but that's OK. If I wanted, I could use a mixture on von Mises distributions.

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