Module « scipy.stats »

Fonction burr - module scipy.stats

Signature de la fonction burr

def burr(*args, **kwds) 

Description

burr.__doc__

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(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)
        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(alpha, c, d, loc=0, scale=1)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    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 x^{-c - 1} / (1 + x^{-c})^{d + 1}

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

    `burr` takes :math:`c` and :math:`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 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
    --------
    >>> 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, histtype='stepfilled', alpha=0.2)
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()