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.. _rst_dist_mauchly: 

Distribution of Mauchly's test of sphericity vs general structure
-------------------------------------------------------------------------------


.. py:class:: ctx.dist_mauchley(p, bi, ci)

    where ``ctx`` is ``fpm``, ``mpm``, ``ipm``, ``dec``, ``gmp`` or ``apm``.

    A random variable `X` follows the distribution of the negative logarithm of the product of `p` beta variables with parameters `a_i` and `b_i` if it is defined as `X = -\log(Y)`, where `Y` follows a beta product distribution  with parameters `a_i` and `b_i`. The  support interval of `X` is `(0,+\infty)`.

    See also :cite:t:`Mauchly1940`,  :cite:t:`Anderson2003`, :cite:t:`Muirhead1982`, :cite:t:`Butler2007`, :cite:t:`Ginzberg2013`, pages 92-105, :cite:t:`Marques2011`, :cite:t:`Tang1984`.




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.. method:: dist_mauchley.pdf(x)

    Returns `\text{pdf}_X(x)`, the probability density function (pdf) of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables:

    The pdf can be calculated (in principle in arbitrary precision) by numerical inversion of the characteristic function, using the algorithm by Gil-Pelaez. The PDF of Y is the inverse Fourier transform of its characteristic function,

    .. math:: \text{pdf}_X(x) = \frac{1}{\pi} \int_{0}^{\infty} \Re \left ( e^{-itx} C_X(t) \right ) \mathrm{d} t.

    where `\Re (z)` denotes the real part of `z`. 


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; x = 3; 
        >>> print ("pdf: ", fisher_f(mu, sigma).pdf(x))
        6.3563523462564525615615615614561356E-20



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.. method:: dist_mauchley.cdf(x)

    Returns `\text{cdf}_X(x)`, the cumulative distribution function (cdf) of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables:


    The cdf can be calculated (in principle in arbitrary precision) by numerical inversion of the characteristic function, using the algorithm by Gil-Pelaez. Gil-Pelaez  derived the following inversion formula which requires integration of a real-valued function, only. In particular,

    .. math:: \text{cdf}_X(x) = \frac{1}{2} - \frac{1}{\pi} \int_{0}^{\infty} \Im \left (    \frac{  e^{-itx} C_X(t)}{t}  \right ) \mathrm{d} t.

    where `\Im (z)` denotes the imaginary part of `z`.


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; x = 3; 
        >>> print ("cdf: ", fisher_f(mu, sigma).pdf(x))
        6.3563523462564525615615615614561356E-20




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.. method:: dist_mauchley.sf(x)

    Returns `\text{sf}_X(x)`, the survival function (sf) of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables:


    The sf can be calculated (in principle in arbitrary precision) by numerical inversion of the characteristic function, using the algorithm by Gil-Pelaez. Gil-Pelaez  derived the following inversion formula which requires integration of a real-valued function, only. In particular,

    .. math:: \text{sf}_X(x) = \frac{1}{2} + \frac{1}{\pi} \int_{0}^{\infty} \Im \left (    \frac{  e^{-itx} C_X(t)}{t}  \right ) \mathrm{d} t.

    where `\Im (z)` denotes the imaginary part of `z`.



    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; x = 3; 
        >>> print (" sf: ", fisher_f(mu, sigma).pdf(x))
        sf: 6.3563523462564525615615615614561356E-20




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.. method:: dist_mauchley.qtf(q)

    Returns `\text{qtf}_X(q)`, the quantile function (qtf) of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables:

    There is no known closed exact form for `\text{qtf}_X(q)` or `\text{isf}_X(q)`. It is computed with Newton iterations
    where the starting values are from Nagarsenker's approximation.


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("qtf: ", fisher_f(mu, sigma).qtf(q))
        qtf: 6.3563523462564525615615615614561356E+00




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.. method:: dist_mauchley.isf(q)

    Returns `\text{isf}_X(q)`, the inverse survival function (isf) of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables:

    There is no known closed exact form for `\text{isf}_X(q)` or `\text{isf}_X(q)`. It is computed with Newton iterations
    where the starting values are from Nagarsenker's approximation.


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", fisher_f(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00




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.. method:: dist_mauchley.c_x(t)

    Returns `C_X(t)`, the characteristic function of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables:


    .. math:: C_X(t) =  \prod_{j=1}^p  \frac{\Gamma\left((a_j-it)\right) \Gamma\left((a_j+b_j)\right)}{\Gamma\left(a_j\right) \Gamma\left(a_j+b_j-it\right)}


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", fisher_f(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00




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.. method:: dist_mauchley.m_x(t)

    Returns the moment generating function of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables.

    .. math:: M_X(t) = \prod_{j=1}^p  \frac{\Gamma\left((a_j-t)\right) \Gamma\left((a_j+b_j)\right)}{\Gamma\left(a_j\right) \Gamma\left(a_j+b_j-t\right)}


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", fisher_f(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00




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.. method:: dist_mauchley.k_x(t, k = 0)

    Returns `K_X(t)`, the cumulant generating function, and its `r^{\text{th}}` derivatives, `K_X^{(r)}(t), r = 1 \ldots k`, of a random variable `X`, following the distribution of the negative logarithm of the product of independent beta variables.

    .. math:: K_X(t) =  \sum_{j=1}^p  \log \left(\Gamma(a_j-t)\right) - \log \left(\Gamma(a_j+b_j-t)\right)  +\log\left(\Gamma(a_j+b_j)\right) -\log\left(\Gamma(a_j)\right).

    .. math:: K^{(r)}_X(t) = (-1)^r \sum_{j=1}^p  \left( \psi^{(r-1)}(a_j-t) - \psi^{(r-1)}(a_j+b_j-t) \right) 

    where `\psi^{(r)}(\cdot)` is the polygamma function of order `r`.


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", fisher_f(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00




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.. method:: dist_mauchley.moments(k)

    Returns the first `j` raw moments, `\mu_j, j = 1 \ldots k`, of a random variable `X`, 
    following the distribution of the negative logarithm of the product of independent beta variables. The moments are calculated from the cumulants.


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", fisher_f(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00



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.. method:: dist_mauchley.cumulants(k)

    Returns the first `j` cumulants, `\kappa_r, r = 1 \ldots k`, of a random variable `X`, 
    following the distribution of the negative logarithm of the product of independent beta variables. 

    .. math:: \kappa_r = (-1)^r  \sum_{j=1}^p  \left( \psi^{(r-1)}(a_j) - \psi^{(r-1)}(a_j+b_j) \right) 

    where `\psi^{(r)}(\cdot)` is the polygamma function of order `r`.


    .. code-block:: python

        >>> from mpfunlab import *
        >>> mp.dps = 30
        >>> mu = 0; sigma = 1; q = 0.3; 
        >>> print ("isf: ", fisher_f(mu, sigma).isf(q))
        6.3563523462564525615615615614561356E+00







**Approximations**



.. method:: ctx.mauchley_bd(x, f, rho, omega)

    where ``ctx`` is ``ipm``, ``dec``, ``mpm``, or ``gmp``.


    The likelihood criterion for testing the hypothesis that a sample of size `N` is drawn from a `p`-variate normal population whose covaraince matrix is proportional to a given matrix `\Sigma_0` is a power of

    .. math:: W = |S \Sigma_0^{-1}|^{\tfrac{1}{2}n} \left(tr(S \Sigma_0^{-1})/p \right)^{\tfrac{1}{2}pn}

    where `n=N-1` and `S` is the sample covariance matrix. We have

    .. math:: f=\tfrac{1}{2}(p-1)(p+2); \quad \rho=1-\frac{2p^2+p+2}{6pn}


    .. math:: \omega_r = \frac{2(-1)^r}{r(r+1)(r+2) \rho^r} \sum_{s=1}^{r+2} \binom{r+2}{s+1} (1-\rho)^{r+1-s} \frac{\delta_s + \tfrac{1}{2}(s+1) B_s / p^{s-1}}{(\tfrac{1}{2}^{s-1})}, 

    where `B_s` are the Bernoulli numbers.



.. method:: ctx.mauchley_bd_inv(q, f, rho, omega)

    where ``ctx`` is ``ipm``, ``dec``, ``mpm``, or ``gmp``.

    Calculates the Box-Davis approximation to the qtf and isf.