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Ok, this takes a little bit o preparation. I'd like to estimate a distribution for some data. I have a distribution function that is not part of the canonical Mathematica set so I define it via ProbabilityDensity

betaPrimeDistribution[ a_, b_] = 
  ProbabilityDistribution[ 
   x^(a - 1) (1 + x) ^(-a - b)/Beta[a, b], {x, 0, ∞}, 
   Assumptions -> a > 0 && b > 0];

If I run EstimatedDistribution on my data

EstimatedDistribution[ data, betaPrimeDistribution[a, b]]

I get the following

ProbabilityDistribution[0.258203/(\[FormalX]^0.68685 (1 + \
\[FormalX])^0.963023), {\[FormalX], 0, ∞}]

So, that is fine. But I have a somewhat largish number of distributions defined via ProbabilityDistribution, so for ease of parsing, what I want to get out is something like

betaPrimeDistribution[0.314, 0.649]

the same way as if I use distributions that are "known" to Mathematica. So, in short, I'd like EstimatedDistribution to return the function name, not the resolved expression.

I tried SetDelayed (:=) in the definition of betaPrimeDistribution - to no avail.

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    $\begingroup$ Perhaps it would be more convenient for you to use FindDistributionParameters[] instead, and plug the parameters generated from that into an Inactive[] version of your custom distribution. $\endgroup$ – J. M.'s ennui Aug 27 '20 at 17:31
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After a bit of spelunking, you could try the following to define a new distribution:

betaPrimeDistributionEval[a_,b_] := ProbabilityDistribution[x^(a-1) (1+x)^(-a-b)/Beta[a,b],{x,0,\[Infinity]},Assumptions->a>0&&b>0]
betaPrimeDistribution /: DistributionDomain[_betaPrimeDistribution] = Interval[{0, Infinity}];
betaPrimeDistribution /: DistributionParameterQ[_betaPrimeDistribution] = True;
betaPrimeDistribution /: DistributionParameterAssumptions[betaPrimeDistribution[a_,b_]] := a>0 && b>0
betaPrimeDistribution /: PDF[betaPrimeDistribution[a_, b_], c___] := PDF[betaPrimeDistributionEval[a, b], c]
betaPrimeDistribution /: RandomVariate[betaPrimeDistribution[a_, b_], c___] := RandomVariate[betaPrimeDistributionEval[a, b], c]

Then I think the following does what you want:

data = RandomVariate[betaPrimeDistribution[1, 2], 100];
EstimatedDistribution[data, betaPrimeDistribution[a, b]]

betaPrimeDistribution[0.899712, 1.704]

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  • $\begingroup$ Ok, cool. That works. I'd be very interested to learn about your "spelunking" and to understand why it works. $\endgroup$ – Oliver Jennrich Aug 28 '20 at 12:30
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What do you mean? The beta prime distribution is right here. Even if it wasn't, you could easily define it with TransformedDistribution as

betaPrimeDistribution[α_, β_] := 
 TransformedDistribution[
   \[FormalX]/(1 - \[FormalX]),
   \[FormalX] \[Distributed] BetaDistribution[α, β]
]
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  • $\begingroup$ Yes. So I used BetaPrime because it was the first one I had in the code. Imagine any other, such like the generalized Beta or any distribution that is exotic enough to not be in Mathematica's canonical set. $\endgroup$ – Oliver Jennrich Aug 28 '20 at 12:34
  • $\begingroup$ @OliverJennrich In that case you should make clear in the question that it's just an example. As it stands, the question simply asks how to fit a beta prime distribution. $\endgroup$ – Sjoerd Smit Aug 28 '20 at 13:51

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