Distribution of Wilks’ \(\Lambda\)

class ctx.dist_wilks_lambda(p, m, n)

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

The Wilks’ \(\Lambda\) distribution is a continuous probability distribution with \(p \ge 1\) predictor variables, error degress of freedom \(m \ge 1\) and \(n \ge 1\), and the support interval \((0,1)\). See also: Wilks [1602], Anderson [9], Muirhead [440], Butler [172], Pham-Gia [494], Witkovský [1649], Witkovský [1648].

dist_wilks_lambda.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following the distribution of Wilks’ Lambda: 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(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_wilks_lambda.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following the distribution of Wilks’ Lambda: 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(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_wilks_lambda.sf(x)

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

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

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

dist_wilks_lambda.qtf(q)

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

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(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_wilks_lambda.isf(q)

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

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

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

dist_wilks_lambda.c_x(t)

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

\[C_X(t) = \frac{\Gamma_p(n/2 -it)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 -it)}.\]

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

dist_wilks_lambda.m_x(t)

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

\[M_X(t) = \frac{\Gamma_p(n/2 + s)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 + s)}.\]

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

dist_wilks_lambda.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 distribution of`2log W` of central Wilks \(W\):

\[K_X(s) = \log \left[ \frac{\Gamma_p(n/2 + s)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 + s)} \right].\]

\[K'(s) = \sum_{i=1}^p \left[\psi \left( \tfrac{1}{2}n+s - \tfrac{1}{2}(i-1)\right) - \psi \left( \tfrac{1}{2}(n+m)+s - \tfrac{1}{2}(i-1)\right) \right]\]

where \(\Gamma_p(\cdot)\) is the multivariate gamma function and \(\psi(\cdot)\) is the digamma function.

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

dist_wilks_lambda.moments(k)

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

\[\mu'_X(r) = \frac{\Gamma_p(n/2 + s)\Gamma_p((n + m)/2)}{\Gamma_p(n/2)\Gamma_p((n + m)/2 + s)}.\]

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

dist_wilks_lambda.cumulants(k)

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

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

Additional information

Anderson, 2003, p. 651-656

p: # of variables

m = q1 = # of groups

M: n-p+1; N=n+q; n = N-q = degrees of freedom (error)

Tables: Renscher 2002, p.566 - 573

Approximations

ctx.wilks_lambda_gp(x, p, m, k, results='cdf')

where ctx is fpm, mpm, ipm, dec, 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_ecf(x, p, m, n, results='cdf')

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

Calculates the Edgeworth approximation to the pdf, cdf and sf.

ctx.wilks_lambda_ecf_inv(q, p, m, n, results='qtf')

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

Calculates the Cornish-Fisher approximation to the qtf and isf.

ctx.wilks_lambda_spa((x, p, n1, n2, 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 \(2\log W\).

ctx.wilks_lambda_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 \(2\log W\).

ctx.wilks_lambda_bd(x, f, rho, omega)

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

Calculates the Box-Davis approximation to the pdf.

For the Wilks’ Lambda distribution, the parameters of the Box-Davis expansion are given vy

\[f=pq; \quad \rho=1-\frac{p+q+1}{2(N-1)},\]

\[\omega_r = \frac{(-2)^r}{r(r+1) \mu^r} \sum_{j=0}^{q-1}{B_{r+1}((\beta -j)/2) - B_{r+1}((\beta -p-j)/2)}, \quad \text{where}\]

\[\beta=\frac{p+q+1}{2}; \quad \mu=N-\frac{p+q+3}{2}.\]

wilks_lambda_bd_inv(q, f, rho, omega)

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

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