Noncentral Distribution of Wilks’ \(\Lambda\): Canonical Correlation

class ctx.dist_wilks_lambda_corr(p1, p2, n, Rho2)

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

The noncentral Wilks’ \(\Lambda\) distribution under the independence alternative, with \(p_1 \ge 1\) and \(p_2 \ge 1\) groups of variables, error degress of freedom \(n \ge 1\), noncentrality parameter \(P^2\) with diagonal entries \(\rho^2_{jj} \in (0,1)\) is a continuous probability distribution with the support interval \((0,1)\). See also Anderson [9], Muirhead [440], Butler [172], Butler and Wood [174], Butler and Wood [175], Fujikoshi [353], Fujikoshi [354], Lee [427], Lee [428], Walster and Tretter [864], Witkovský [1634].

dist_wilks_lambda_corr.pdf(x, kwargs)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR): The pdf is computed by numerical inversion of the characteristic function or cumulant generating function.

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

dist_wilks_lambda_corr.cdf(x, kwargs)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR): The cdf is computed by numerical inversion of the characteristic function or cumulant generating function.

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

dist_wilks_lambda_corr.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

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

dist_wilks_lambda_corr.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

There is no known closed form for the quantile function \(\text{cdf}^{-1}_X(q)\): It is computed with Newton iterations where the starting values are from a central chi-square approximation.

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

dist_wilks_lambda_corr.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

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

dist_wilks_lambda_corr.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

\[C_X(t) = \frac{\Gamma_{p_1}(n/2)\Gamma_{p_1}((n - p_2)/2 -it)}{\Gamma_{p_1}(n/2 -it)\Gamma_{p_1}((n-p_2)/2)} \times \vert I_{p_1}-P^2 \vert ^{n/2} {}_2F_1\left(\frac{n}{2},\frac{n}{2} ;\frac{n}{2} -it ; P^2\right).\]

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

dist_wilks_lambda_corr.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

\[M_X(t) = \left[ \frac{\Gamma_{p_1}(n/2)\Gamma_{p_1}((n - p_2)/2 + s)}{\Gamma_{p_1}(n/2+s)\Gamma_{p_1}((n-p_2)/2)} \times \vert I_{p_1}-P^2 \vert ^{n/2} {}_2F_1\left(\frac{n}{2},\frac{n}{2} ;\frac{n}{2}+ s ; P^2\right) \right].\]

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

dist_wilks_lambda_corr.k_x(s, k=0)

Returns \(K_X(s)\), the cumulant generating function, and its \(j^{\text{th}}\) derivatives, \(K_X^{(j)}(s), j = 1 \ldots k\), of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

\[K_X(s) = \log \left[ \frac{\Gamma_{p_1}(n/2)\Gamma_{p_1}((n - p_2)/2 + s)}{\Gamma_{p_1}(n/2+s)\Gamma_{p_1}((n-p_2)/2)} \times \vert I_{p_1}-P^2 \vert ^{n/2} {}_2F_1\left(\frac{n}{2},\frac{n}{2} ;\frac{n}{2}+ s ; P^2\right) \right].\]

\begin{eqnarray} K'(s) & = & \sum_{i=1}^p \left[\psi \left( \tfrac{1}{2}n+s - \tfrac{1}{2}(i-1)\right) - \psi \left( \tfrac{1}{2}(n+m)+s - \tfrac{1}{2}(i-1)\right) \right] \nonumber \\ & + & \frac{\partial}{\partial s} \log \left[ \vert I_{p_1}-P^2 \vert ^{n/2} {}_2F_1\left(\frac{n}{2},\frac{n}{2} ;\frac{n}{2}+ s ; P^2\right) \right] \nonumber , \end{eqnarray}

where \(\Gamma_p(\cdot)\) is the multivariate gamma function , \(\psi(\cdot)\) is the digamma function , and \({}_2F_1(\cdot,\cdot,X)\) is the Gauss hypergeometric function of matrix argument . The saddlepoint equation needs to be evaluated numerically. Also, the computation of \(K''(s)\) is performed using a numerical derivative of \(K'(s)\).

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", wilks_lambda_corr(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_wilks_lambda_corr.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR):

\[\mu'_X(r) = \frac{\Gamma_{p_1}(n/2)\Gamma_{p_1}((n - p_2)/2 + s)}{\Gamma_{p_1}(n/2+s)\Gamma_{p_1}((n-p_2)/2)} \times \vert I_{p_1}-P^2 \vert ^{n/2} {}_2F_1\left(\frac{n}{2},\frac{n}{2} ;\frac{n}{2}+ s ; P^2\right).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", wilks_lambda_corr(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_wilks_lambda_corr.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following the noncentral distribution of Wilks’ Lambda (CORR) The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", wilks_lambda_corr(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00

Approximations

ctx.wilks_lambda_ind_gp(x, p1, p2, n, Rho2, method='default')

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

Calculates the pdf, cdf and sf from the characteristic function using the procedure of Gil-Pelaez (see gil_pelaez_pdf() and gil_pelaez_cdf()).

This uses \(U = \log 2 W\).

ctx.wilks_lambda_ind_spa(x, n, results='c')

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

Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf.

This uses \(2\log W\) of non-central Wilks \(W\)

ctx.wilks_lambda_ind_spa_inv(x, n, results='qtf')

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

Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.

This uses \(2\log W\) of non-central Wilks \(W\)