scipy.stats.burr12

scipy.stats.burr12 = <scipy.stats._continuous_distns.burr12_gen object>

A Burr (Type XII) continuous random variable.

As an instance of the rv_continuous class, burr12 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.

Methods

rvs(c, d, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, c, d, loc=0, scale=1)	Probability density function.
logpdf(x, c, d, loc=0, scale=1)	Log of the probability density function.
cdf(x, c, d, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, c, d, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, c, d, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, c, d, loc=0, scale=1)	Log of the survival function.
ppf(q, c, d, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, c, d, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(order, c, d, loc=0, scale=1)	Non-central moment of the specified order.
stats(c, d, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(c, d, loc=0, scale=1)	(Differential) entropy of the RV.
fit(data)	Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
expect(func, args=(c, d), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds)	Expected value of a function (of one argument) with respect to the distribution.
median(c, d, loc=0, scale=1)	Median of the distribution.
mean(c, d, loc=0, scale=1)	Mean of the distribution.
var(c, d, loc=0, scale=1)	Variance of the distribution.
std(c, d, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, c, d, loc=0, scale=1)	Confidence interval with equal areas around the median.

See also

fisk

a special case of either burr or burr12 with d=1

burr

Burr Type III distribution

Notes

The probability density function for burr12 is:

\[f(x; c, d) = c d \frac{x^{c-1}} {(1 + x^c)^{d + 1}}\]

for \(x >= 0\) and \(c, d > 0\).

burr12 takes c and d as shape parameters for \(c\) and \(d\).

This is the PDF corresponding to the twelfth CDF given in Burr’s list; specifically, it is equation (20) in Burr’s paper [1].

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, burr12.pdf(x, c, d, loc, scale) is identically equivalent to burr12.pdf(y, c, d) / scale with y = (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.

The Burr type 12 distribution is also sometimes referred to as the Singh-Maddala distribution from NIST [2].

References

[1]

Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942).

[2]

https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/b12pdf.htm

[3]

“Burr distribution”, https://en.wikipedia.org/wiki/Burr_distribution

Examples

Try it in your browser!

>>> import numpy as np
>>> from scipy.stats import burr12
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> c, d = 10, 4
>>> mean, var, skew, kurt = burr12.stats(c, d, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(burr12.ppf(0.01, c, d),
...                 burr12.ppf(0.99, c, d), 100)
>>> ax.plot(x, burr12.pdf(x, c, d),
...        'r-', lw=5, alpha=0.6, label='burr12 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 = burr12(c, d)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = burr12.ppf([0.001, 0.5, 0.999], c, d)
>>> np.allclose([0.001, 0.5, 0.999], burr12.cdf(vals, c, d))
True

Generate random numbers:

>>> r = burr12.rvs(c, d, 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()

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