Distribution of Mauchly’s test of sphericity vs general structure

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 Mauchly [434], Anderson [9], Muirhead [440], Butler [172], Ginzberg [364], pages 92-105, Marques et al. [433], Tang and Gupta [539].

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,

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

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

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

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,

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

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

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

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,

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

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

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

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.

>>> 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

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.

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

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:

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

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

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.

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

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

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.

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

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

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

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

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.

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

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.

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

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

>>> 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

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

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

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

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

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

where \(B_s\) are the Bernoulli numbers.

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.