How to generate a distribution with a given mean, variance, skew and kurtosis in Python?

Question

random.gauss(mu, sigma)

Above is a function allowing to randomly draw a number from a normal distribution with a given mean and variance. But how can we draw values from a normal distribution defined by more than only the two first moments?

something like:

random.gauss(mu, sigma, skew, kurtosis)

Solution 1

How about using scipy? You can pick the distribution you want from continuous distributions in the scipy.stats library.

The generalized gamma function has non-zero skew and kurtosis, but you'll have a little work to do to figure out what parameters to use to specify the distribution to get a particular mean, variance, skew and kurtosis. Here's some code to get you started.

import scipy.stats
import matplotlib.pyplot as plt
distribution = scipy.stats.norm(loc=100,scale=5)
sample = distribution.rvs(size=10000)
plt.hist(sample)
plt.show()
print distribution.stats('mvsk')

This displays a histogram of a 10,000 element sample from a normal distribution with mean 100 and variance 25, and prints the distribution's statistics:

(array(100.0), array(25.0), array(0.0), array(0.0))

Replacing the normal distribution with the generalized gamma distribution,

distribution = scipy.stats.gengamma(100, 70, loc=50, scale=10)

you get the statistics [mean, variance, skew, kurtosis] (array(60.67925117494595), array(0.00023388203873597746), array(-0.09588807605341435), array(-0.028177799805207737)).

Solution 2

Try to use this:

http://statsmodels.sourceforge.net/devel/generated/statsmodels.sandbox.distributions.extras.pdf_mvsk.html#statsmodels.sandbox.distributions.extras.pdf_mvsk

Return the Gaussian expanded pdf function given the list of 1st, 2nd moment and skew and Fisher (excess) kurtosis.

Parameters : mvsk : list of mu, mc2, skew, kurt

Looks good to me. There's a link to the source on that page.

Oh, and here's the other StackOverflow question that pointed me there: Apply kurtosis to a distribution in python

Comments

[Bennett Brown]

Any normal distribution has skew 0 and kurtosis 0. Use a different family of distributions.

[Bennett Brown]

Beware, there are several ways to define the calculation for skew and kurtosis. Moments are not equivalent to mean, variance, skew, and kurtosis, though they have the same gist.

[Bennett Brown]

Also, the moments do not specify a unique distribution. See [this similar question but asking about R: stackoverflow.com/questions/4807398/…