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When searching for the best-fit distribution for my dataset, the result was the Exponentially Modified Normal distribution with the following parameters: 

K=10.84, loc=154.35, scale=73.82

Scipy gives us a way to analyze the mean of the distribution by:

  fitted_mean =  scipy.stats.exponnorm.stats(K=10.84, loc=154.35, scale=73.82, moments='mean')

The resulting fitted_mean=984, which is the same mean as my dataset. However, I'm not sure what this is telling me. I thought the loc=154.35 is the mean of the distribution.

What are these two means? If I fitted the data with the best distribution, isn't the fitted_mean (154.35) the new and only mean?

Adam Schroeder
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1 Answers1

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For the exponentially modified normal distribution, the location parameter is not the same as the mean. This is true for many distributions.

Take a look at the wikipedia page for the exponentially modified Gaussian distribution. This is the same distribution as scipy.stats.exponnorm, but with a different parameterization. The mapping of the parameters between the wikipedia version and scipy is:

μ = loc
σ = scale
λ = 1/(K*scale)

The wikipedia page says the mean of the distribution is μ + 1/λ, which, in terms of the scipy parameters, is loc + K*scale.

When you fit the distribution to your data, you found

loc = 154.35
scale = 73.82 
K = 10.84

The formula for the mean from the wikipedia page gives

loc + K*scale = 954.5587999999999

Here is the calculation using exponnorm:

In [16]: fitted_mean = scipy.stats.exponnorm.stats(K=10.84, loc=154.35, scale=73.82, moments='mean')

In [17]: fitted_mean
Out[17]: array(954.5587999999999)

which matches the result from the wikipedia formula.

(You reported fitted_mean = 984, but I assume that was a typographical error.)

Warren Weckesser
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  • Thank you for clarifying @Warren Weckesser . Yes. it was a typo. So the loc parameter in the exponnonm distribution is just a location parameter, not the mean? But in other distributions like the normal distribution it is the mean? – Adam Schroeder Mar 21 '18 at 03:55
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    Yes, exactly. Another example is the [Gumbel distribution](https://en.wikipedia.org/wiki/Gumbel_distribution): μ is the location parameter, but the mean is μ + βγ, where β is the scale parameter and γ is Euler-Mascheroni constant. – Warren Weckesser Mar 21 '18 at 04:21