scipy.stats.kappa4 = <scipy.stats._continuous_distns.kappa4_gen object>[source]#

Kappa 4 parameter distribution.

As an instance of the rv_continuous class, kappa4 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 kappa4 is:

\[f(x, h, k) = (1 - k x)^{1/k - 1} (1 - h (1 - k x)^{1/k})^{1/h-1}\]

if \(h\) and \(k\) are not equal to 0.

If \(h\) or \(k\) are zero then the pdf can be simplified:

h = 0 and k != 0:

kappa4.pdf(x, h, k) = (1.0 - k*x)**(1.0/k - 1.0)*
                      exp(-(1.0 - k*x)**(1.0/k))

h != 0 and k = 0:

kappa4.pdf(x, h, k) = exp(-x)*(1.0 - h*exp(-x))**(1.0/h - 1.0)

h = 0 and k = 0:

kappa4.pdf(x, h, k) = exp(-x)*exp(-exp(-x))

kappa4 takes \(h\) and \(k\) as shape parameters.

The kappa4 distribution returns other distributions when certain \(h\) and \(k\) values are used.








Generalized Logistic(1)




Reverse Exponential(2)

Generalized Extreme Value

genextreme(x, k)






Generalized Pareto

genpareto(x, -k)

  1. There are at least five generalized logistic distributions. Four are described here: The “fifth” one is the one kappa4 should match which currently isn’t implemented in scipy:

  2. This distribution is currently not in scipy.


J.C. Finney, “Optimization of a Skewed Logistic Distribution With Respect to the Kolmogorov-Smirnov Test”, A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College, (August, 2004),

J.R.M. Hosking, “The four-parameter kappa distribution”. IBM J. Res. Develop. 38 (3), 25 1-258 (1994).

B. Kumphon, A. Kaew-Man, P. Seenoi, “A Rainfall Distribution for the Lampao Site in the Chi River Basin, Thailand”, Journal of Water Resource and Protection, vol. 4, 866-869, (2012). DOI:10.4236/jwarp.2012.410101

C. Winchester, “On Estimation of the Four-Parameter Kappa Distribution”, A Thesis Submitted to Dalhousie University, Halifax, Nova Scotia, (March 2000).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, kappa4.pdf(x, h, k, loc, scale) is identically equivalent to kappa4.pdf(y, h, k) / 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.


>>> import numpy as np
>>> from scipy.stats import kappa4
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> h, k = 0.1, 0
>>> mean, var, skew, kurt = kappa4.stats(h, k, moments='mvsk')

Display the probability density function (pdf):

>>> x = np.linspace(kappa4.ppf(0.01, h, k),
...                 kappa4.ppf(0.99, h, k), 100)
>>> ax.plot(x, kappa4.pdf(x, h, k),
...        'r-', lw=5, alpha=0.6, label='kappa4 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 = kappa4(h, k)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

>>> vals = kappa4.ppf([0.001, 0.5, 0.999], h, k)
>>> np.allclose([0.001, 0.5, 0.999], kappa4.cdf(vals, h, k))

Generate random numbers:

>>> r = kappa4.rvs(h, k, 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(h, k, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, h, k, loc=0, scale=1)

Probability density function.

logpdf(x, h, k, loc=0, scale=1)

Log of the probability density function.

cdf(x, h, k, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, h, k, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, h, k, loc=0, scale=1)

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

logsf(x, h, k, loc=0, scale=1)

Log of the survival function.

ppf(q, h, k, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, h, k, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, h, k, loc=0, scale=1)

Non-central moment of the specified order.

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

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

entropy(h, k, 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=(h, k), 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(h, k, loc=0, scale=1)

Median of the distribution.

mean(h, k, loc=0, scale=1)

Mean of the distribution.

var(h, k, loc=0, scale=1)

Variance of the distribution.

std(h, k, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, h, k, loc=0, scale=1)

Confidence interval with equal areas around the median.