scipy.stats.gengamma — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.gengamma = <scipy.stats._continuous_distns.gengamma_gen object>[source]#

A generalized gamma continuous random variable.

As an instance of the rv_continuous class, gengamma object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Notes

The probability density function for gengamma is ([1]):

\[f(x, a, c) = \frac{|c| x^{c a-1} \exp(-x^c)}{\Gamma(a)}\]

for \(x \ge 0\), \(a > 0\), and \(c \ne 0\).\(\Gamma\) is the gamma function (scipy.special.gamma).

gengamma takes \(a\) and \(c\) as shape parameters.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, gengamma.pdf(x, a, c, loc, scale) is identically equivalent to gengamma.pdf(y, a, c) / scale withy = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

References

[1]

E.W. Stacy, “A Generalization of the Gamma Distribution”, Annals of Mathematical Statistics, Vol 33(3), pp. 1187–1192.

Examples

import numpy as np from scipy.stats import gengamma import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

a, c = 4.42, -3.12 mean, var, skew, kurt = gengamma.stats(a, c, moments='mvsk')

Display the probability density function (pdf):

x = np.linspace(gengamma.ppf(0.01, a, c), ... gengamma.ppf(0.99, a, c), 100) ax.plot(x, gengamma.pdf(x, a, c), ... 'r-', lw=5, alpha=0.6, label='gengamma pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

rv = gengamma(a, c) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

vals = gengamma.ppf([0.001, 0.5, 0.999], a, c) np.allclose([0.001, 0.5, 0.999], gengamma.cdf(vals, a, c)) True

Generate random numbers:

r = gengamma.rvs(a, c, size=1000)

And compare the histogram:

ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) ax.set_xlim([x[0], x[-1]]) ax.legend(loc='best', frameon=False) plt.show()

Methods