Parameter Estimation of a mixture of sums of Lognormals

Question

Let $X$ be a random variable that is of a mixture distribution of two lognormals (with the same sigma), so $X \approx p\cdot \mathcal L (\mu_1, \sigma) + (1-p)\cdot \mathcal L (\mu_2,\sigma)$.

Now set $Y=X_1 + X_2$, so $Y$ is a sum of two samples from this mixture.

First I generate some data according to this distribution:

dMix[p_, m1_, m2_, s_] := 
MixtureDistribution[{p, 1 - p}, {LogNormalDistribution[m1, s], 
LogNormalDistribution[m2, s]}];

dat = Plus @@ RandomVariate[dMix[0.75, 0.5, -1.5, 0.2], {2,100}];

this is the pdf of my distribution, using TransformedDistribution

dMixSum = 
 TransformedDistribution[x1 + x2, 
  Thread[Distributed[{x1, x2}, 
   MixtureDistribution[{p, 1 - p}, {LogNormalDistribution[m1, s], 
    LogNormalDistribution[m2, s]}]]]];

pdf[z_] = PDF[dMixSum, z];

pDist = ProbabilityDistribution[pdf[z], {z, -Infinity, Infinity}, 
  Assumptions -> DistributionParameterAssumptions[dMixSum]]

Now FindDistributionParameters should do the job (but it doesn't)

FindDistributionParameters[dat, pDist, {{m1, 1}, {m2, -1}, {s, 
 1}, {p, .5}}]

I think this is because the sum of two lognormals is not lognormally distributed (in a closed form), it is only approximately lognormal. Can I tell mathematica somehow to take this into account or is there another way to solve this?

Answer 1

This is somewhere between a comment and an answer, but here's a histogram of your sum distribution:

dMix[p_, m1_, m2_, s_] := 
  MixtureDistribution[{p, 1 - p}, {LogNormalDistribution[m1, s], 
    LogNormalDistribution[m2, s]}];
Histogram[
 Plus @@ RandomVariate[dMix[0.75, 0.5, -1.5, 0.2], {2, 400000}]]

Meanwhile, here is a histogram of one of a typical lognormal distribution (using the same $m_1,s$ parameters you specified):

Histogram@RandomVariate[LogNormalDistribution[0.5, 0.2], 200000]

The sum distribution $Y$ you specified does not remotely look like a lognormal distribution (for example, it's trimodal!), so I'm not really sure if it even makes sense to do a parameter distribution on $Y$ by assuming that it's approximately lognormal, since that approximation does not appear to be valid.