scipy.stats.invweibull = <scipy.stats._continuous_distns.invweibull_gen object>[source]#

An inverted Weibull continuous random variable.

This distribution is also known as the Fréchet distribution or the type II extreme value distribution.

As an instance of the rv_continuous class, invweibull 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.


The probability density function for invweibull is:

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

for \(x > 0\), \(c > 0\).

invweibull takes c as a shape parameter for \(c\).

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


F.R.S. de Gusmao, E.M.M Ortega and G.M. Cordeiro, “The generalized inverse Weibull distribution”, Stat. Papers, vol. 52, pp. 591-619, 2011.


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

Calculate the first four moments:

>>> c = 10.6
>>> mean, var, skew, kurt = invweibull.stats(c, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = invweibull.rvs(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)


rvs(c, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, c, loc=0, scale=1)

Probability density function.

logpdf(x, c, loc=0, scale=1)

Log of the probability density function.

cdf(x, c, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, c, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, c, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, c, loc=0, scale=1)

Log of the survival function.

ppf(q, c, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, c, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, c, loc=0, scale=1)

Non-central moment of the specified order.

stats(c, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(c, loc=0, scale=1)

(Differential) entropy of the RV.


Parameter estimates for generic data. See for detailed documentation of the keyword arguments.

expect(func, args=(c,), 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, loc=0, scale=1)

Median of the distribution.

mean(c, loc=0, scale=1)

Mean of the distribution.

var(c, loc=0, scale=1)

Variance of the distribution.

std(c, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, c, loc=0, scale=1)

Confidence interval with equal areas around the median.