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Fonction studentized_range - module scipy.stats

Signature de la fonction studentized_range

def studentized_range(*args, **kwds) 

Description

studentized_range.__doc__

A studentized range continuous random variable.

    As an instance of the `rv_continuous` class, `studentized_range` 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(k, df, loc=0, scale=1, size=1, random_state=None)
        Random variates.
    pdf(x, k, df, loc=0, scale=1)
        Probability density function.
    logpdf(x, k, df, loc=0, scale=1)
        Log of the probability density function.
    cdf(x, k, df, loc=0, scale=1)
        Cumulative distribution function.
    logcdf(x, k, df, loc=0, scale=1)
        Log of the cumulative distribution function.
    sf(x, k, df, loc=0, scale=1)
        Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
    logsf(x, k, df, loc=0, scale=1)
        Log of the survival function.
    ppf(q, k, df, loc=0, scale=1)
        Percent point function (inverse of ``cdf`` --- percentiles).
    isf(q, k, df, loc=0, scale=1)
        Inverse survival function (inverse of ``sf``).
    moment(n, k, df, loc=0, scale=1)
        Non-central moment of order n
    stats(k, df, loc=0, scale=1, moments='mv')
        Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
    entropy(k, df, 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=(k, df), 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, df, loc=0, scale=1)
        Median of the distribution.
    mean(k, df, loc=0, scale=1)
        Mean of the distribution.
    var(k, df, loc=0, scale=1)
        Variance of the distribution.
    std(k, df, loc=0, scale=1)
        Standard deviation of the distribution.
    interval(alpha, k, df, loc=0, scale=1)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    See Also
    --------
    t: Student's t distribution

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

    .. math::

         f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2)
                        2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty}
                        s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z)
                        [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds

    for :math:`x ≥ 0`, :math:`k > 1`, and :math:`\nu > 0`.

    `studentized_range` takes ``k`` for :math:`k` and ``df`` for :math:`\nu`
    as shape parameters.

    When :math:`\nu` exceeds 100,000, an asymptotic approximation (infinite
    degrees of freedom) is used to compute the cumulative distribution
    function [4]_.

    The probability density above is defined in the "standardized" form. To shift
    and/or scale the distribution use the ``loc`` and ``scale`` parameters.
    Specifically, ``studentized_range.pdf(x, k, df, loc, scale)`` is identically
    equivalent to ``studentized_range.pdf(y, k, df) / 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] "Studentized range distribution",
           https://en.wikipedia.org/wiki/Studentized_range_distribution
    .. [2] Batista, Ben Dêivide, et al. "Externally Studentized Normal Midrange
           Distribution." Ciência e Agrotecnologia, vol. 41, no. 4, 2017, pp.
           378-389., doi:10.1590/1413-70542017414047716.
    .. [3] Harter, H. Leon. "Tables of Range and Studentized Range." The Annals
           of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147.
           JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021.
    .. [4] Lund, R. E., and J. R. Lund. "Algorithm AS 190: Probabilities and
           Upper Quantiles for the Studentized Range." Journal of the Royal
           Statistical Society. Series C (Applied Statistics), vol. 32, no. 2,
           1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18
           Feb. 2021.

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

    Calculate the first four moments:

    >>> k, df = 3, 10
    >>> mean, var, skew, kurt = studentized_range.stats(k, df, moments='mvsk')

    Display the probability density function (``pdf``):

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

    Check accuracy of ``cdf`` and ``ppf``:

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

    Rather than using (``studentized_range.rvs``) to generate random variates,
    which is very slow for this distribution, we can approximate the inverse
    CDF using an interpolator, and then perform inverse transform sampling
    with this approximate inverse CDF.

    This distribution has an infinite but thin right tail, so we focus our
    attention on the leftmost 99.9 percent.

    >>> a, b = studentized_range.ppf([0, .999], k, df)
    >>> a, b
    0, 7.41058083802274

    >>> from scipy.interpolate import interp1d
    >>> rng = np.random.default_rng()
    >>> xs = np.linspace(a, b, 50)
    >>> cdf = studentized_range.cdf(xs, k, df)
    # Create an interpolant of the inverse CDF
    >>> ppf = interp1d(cdf, xs, fill_value='extrapolate')
    # Perform inverse transform sampling using the interpolant
    >>> r = ppf(rng.uniform(size=1000))

    And compare the histogram:

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