Central distribution of Pillai’s \(V\)#

class ctx.dist_pillai_v(p, m, n)#

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

The Pillai \(V\) 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 Anderson [9], Muirhead [440], Butler [172], Davis [209], Davis [210], Davis [211].

dist_pillai_v.pdf(x)#

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following the distribution of Pillai’s V:

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: ", pillai_v(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_pillai_v.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following the distribution of Pillai’s V:

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: ", pillai_v(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_pillai_v.sf(x)#

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

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

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

dist_pillai_v.qtf(q)#

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

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

dist_pillai_v.isf(q)#

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

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

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

dist_pillai_v.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following the distribution of Pillai’s V:

TO BE CALCULATED VIA RAW MOMENTS

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

dist_pillai_v.m_x(t)#

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

TO BE CALCULATED VIA RAW MOMENTS

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

dist_pillai_v.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 Pillai’s V:

TO BE CALCULATED VIA RAW MOMENTS

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

dist_pillai_v.moments(k)#

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

TO BE CALCULATED AS IN DAVIS

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

dist_pillai_v.cumulants(k)#

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

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

Approximations

ctx.pillai_v_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.pillai_v_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.pillai_v_bd(x, f, rho, omega)#

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

Calculates the Box-Davis approximation to the pdf, cdf and sf.

For Pillai’s \(V\) distribution, the parameters of the Box-Davis expansion are given by

Let \(s=1, \quad k=m+1, \quad a=2k+n_1\). Then

\[f=pq; \quad \rho=1.\]

\[\omega_1 = mn_1 k/(2n_2), \quad 2r\omega_r = 2(r-1)\omega_r-1 - s(1-k/n_2)c_{1,r}, \quad r=2,3,...\]

\[c_{0,0} =1; \quad c_{0,r} = c_{r,0} = 0; \quad (r=1,2,...); \quad c_{r,1} = 0 (r=2,3,...)\]

\begin{eqnarray} jc_{j,r} &=& [(m-j+1)(n_1-j+1)]c_{j-1,r-1} + [(j(2m+n_1-2j+2)+2(r-1))/n2]c_{j,r-1} \nonumber \\ &+& [(j+1)/n_2 - (j+1)(m-j+1)/n_2^2]c_{j+1,r-1} -[(mn_1+2(r-2)/n_2]c_{j,r-2} \nonumber \\ &+& (2/n_2) \sum_{i=1}^{r-2} i\omega_i(c_{j,r-i-1} - c_{j,r-i-2}) \nonumber \end{eqnarray}

ctx.pillai_v_bd_inv(q, f, rho, omega)#

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

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

Central distribution of Pillai’s V

Contents

Central distribution of Pillai’s \(V\)#