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Module « scipy.stats »

Fonction burr - module scipy.stats

Signature de la fonction burr

def burr(*args, **kwds) 

Description

help(scipy.stats.burr)

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.

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 <https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.rv_continuous.fit.html#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``
burr12 : Burr Type XII distribution
mielke : Mielke Beta-Kappa / Dagum distribution

Notes
-----
The probability density function for `burr` is:

.. math::

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

for :math:`x >= 0` and :math:`c, d > 0`.

`burr` takes ``c`` and ``d`` as shape parameters for :math:`c` and
:math:`d`.

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 distribution
is also commonly referred to as the Dagum distribution [2]_. If the
parameter :math:`c < 1` then the mean of the distribution does not
exist and if :math:`c < 2` the variance does not exist [2]_.
The PDF is finite at the left endpoint :math:`x = 0` if :math:`c * d >= 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``. 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.

References
----------
.. [1] Burr, I. W. "Cumulative frequency functions", Annals of
   Mathematical Statistics, 13(2), pp 215-232 (1942).
.. [2] https://en.wikipedia.org/wiki/Dagum_distribution
.. [3] Kleiber, Christian. "A guide to the Dagum distributions."
   Modeling Income Distributions and Lorenz Curves  pp 97-117 (2008).

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

Calculate the first four 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, 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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