SciPy

scipy.stats.burr

scipy.stats.burr(*args, **kwds) = <scipy.stats._continuous_distns.burr_gen object>[source]

A Burr (Type III) continuous random variable.

As an instance of the rv_continuous class, burr 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.

See also

fisk

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

burr12

Burr Type XII distribution

Notes

The probability density function for burr is:

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

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

burr takes \(c\) and \(d\) as shape parameters.

This is the PDF corresponding to the third CDF given in Burr’s list; specifically, it is equation (11) 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, burr.pdf(x, c, d, loc, scale) is identically equivalent to burr.pdf(y, c, d) / scale with y = (x - loc) / scale.

References

1

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

Examples

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

Calculate a few first moments:

>>> c, d = 10.5, 4.3
>>> mean, var, skew, kurt = burr.stats(c, d, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = burr.rvs(c, d, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()
../_images/scipy-stats-burr-1.png

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(n, c, d, loc=0, scale=1)

Non-central moment of order n

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, c, d, loc=0, scale=1)

Parameter estimates for generic data.

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(alpha, c, d, loc=0, scale=1)

Endpoints of the range that contains alpha percent of the distribution

Previous topic

scipy.stats.bradford

Next topic

scipy.stats.burr12