# Parameter Estimation of a mixture of sums of Lognormals

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,
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?

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.

• thanks for your comment. I suggest you change LogNormalDistribution to NormalDistribution and you will see that even if the sum is trimodal it will the optimized to a regular mixture of two. If Y=x1+x2+x2 it will have 4 "bumps" and still optimize to a regular mixture of two. This is about finding the global maximum and not a local. – spore234 Aug 4 '14 at 5:57