scipy.stats.mielke

scipy.stats.mielke(*args, **kwds) = <scipy.stats._continuous_distns.mielke_gen object>

A Mielke Beta-Kappa / Dagum continuous random variable.

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

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

for \(x > 0\) and \(k, s > 0\). The distribution is sometimes called Dagum distribution ([2]). It was already defined in [3], called a Burr Type III distribution (burr with parameters c=s and d=k/s).

mielke takes k and s 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, mielke.pdf(x, k, s, loc, scale) is identically equivalent to mielke.pdf(y, k, s) / scale with y = (x - loc) / scale.

References

1

Mielke, P.W., 1973 “Another Family of Distributions for Describing and Analyzing Precipitation Data.” J. Appl. Meteor., 12, 275-280

2

Dagum, C., 1977 “A new model for personal income distribution.” Economie Appliquee, 33, 327-367.

3

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

Examples

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

Calculate a few first moments:

>>> k, s = 10.4, 4.6
>>> mean, var, skew, kurt = mielke.stats(k, s, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

>>> vals = mielke.ppf([0.001, 0.5, 0.999], k, s)
>>> np.allclose([0.001, 0.5, 0.999], mielke.cdf(vals, k, s))
True

Generate random numbers:

>>> r = mielke.rvs(k, s, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Methods

rvs(k, s, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, k, s, loc=0, scale=1)	Probability density function.
logpdf(x, k, s, loc=0, scale=1)	Log of the probability density function.
cdf(x, k, s, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, k, s, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, k, s, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, k, s, loc=0, scale=1)	Log of the survival function.
ppf(q, k, s, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, k, s, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(n, k, s, loc=0, scale=1)	Non-central moment of order n
stats(k, s, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(k, s, loc=0, scale=1)	(Differential) entropy of the RV.
fit(data, k, s, loc=0, scale=1)	Parameter estimates for generic data.
expect(func, args=(k, s), 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(k, s, loc=0, scale=1)	Median of the distribution.
mean(k, s, loc=0, scale=1)	Mean of the distribution.
var(k, s, loc=0, scale=1)	Variance of the distribution.
std(k, s, loc=0, scale=1)	Standard deviation of the distribution.
interval(alpha, k, s, loc=0, scale=1)	Endpoints of the range that contains alpha percent of the distribution