# FindDistributionParameters of a sum of a mixture distribution

I have this Mixture Distribution of two Normal Distributions:

 mix[p_, μ1_, μ2_, σ_] :=
MixtureDistribution[{p,
1 - p}, {NormalDistribution[μ1, σ],
NormalDistribution[μ2, σ]}];


Say $x_1, x_2$ are samples from this mixture. Set $y=x_1+x_2$, so that $y$ is the sum of two samples from the mixture. I now generate 100 $y's$ (with arbitrary p, mu and sigma, each is a sum of two $x$:

mixdat = RandomVariate[mix[0.75, 0.5, -1.5, 0.2], 1000];
mixdatSum =
Total[RandomVariate[mix[0.75, 0.5, -1.5, 0.2], 2]] & /@ Range;


mixdatSum contains these 100 values. Now given the data in mixdatSum I want to estimate the Distribution Parameters. So a natural way to solve this would be:

FindDistributionParameters[mixdatSum,
TransformedDistribution[x1 + x2,
MixtureDistribution[{p,
1 - p}, {NormalDistribution[μ1, σ],
NormalDistribution[μ2, σ]}]]]]]


Problem is, the sum of a mixture does not follow the mixture distribution, so this does not work and I get lots of errors.

However I want to treat the sum of a mixture AS IF IT WERE following the same mixture, for approximation purposes.

Is there a way to do this?

Since the PDF can be computed in closed form you might have some luck with ProbabilityDistribution and some half-way reasonable starting values.

Generate the data...

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

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


Use your TransformedDistribution to get the PDF.

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

pdf[z_] = FullSimplify[PDF[sumdist, z], DistributionParameterAssumptions[sumdist]]


Create a ProbabilityDistribution from your PDF with the proper assumptions set.

pdist = ProbabilityDistribution[pdf[z], {z, -Infinity, Infinity},
Assumptions -> DistributionParameterAssumptions[sumdist]]


With some starting values in the right ballpark you can get good estimates of the parameters.

FindDistributionParameters[mixdatSum, pdist, {{m1, 1}, {m2, -1}, {s, 1}, {p, .5}}]

(*{m1 -> 0.514195, m2 -> -1.53428, s -> 0.226609, p -> 0.719953}

• thanks, this is looking great. Is there a way to get this to work with a Mixture Distribution of two LogNormal Distributions? There is no closed form there. – spore234 Jul 28 '14 at 21:05
• To my knowledge you won't have any luck with ProbabilityDistribution if you don't have a closed form for one of the distribution functions. – Andy Ross Jul 29 '14 at 2:21
• According to a paper I read the sum of two Lognormals is approximately Lognormal. Can I use this to define my own ProbabilityDensity for the sum of two Lognormals? – spore234 Jul 29 '14 at 5:30