Central distribution of Roy’s largest root

Central distribution of Roy’s largest root#

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

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

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following the distribution of Roy’s largest root, with parameters \(p\), \(m\) and \(n\), and the support interval \((0,1)\). See also Anderson [9], Muirhead [440], Butler [172], Chiani [184], Chiani [185].

dist_roy.pdf(x)#

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

\[\text{pdf}_X(x) = \tfrac{1}{2} C \sqrt{|A(\theta_1)|} \: \text{tr} \left( A(t)^{-1} B \right).\]

\(A\) and \(C\) are defined as in roy_pdf_cdf_sf().

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

dist_roy.cdf(x)#

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

\[\text{cdf}_X(x) = C \sqrt{|A(\theta_1)|}.\]

\(A\) and \(C\) are defined as in roy_pdf_cdf_sf().

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

dist_roy.sf(x)#

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

\[\text{sf}_X(x) = 1 - C \sqrt{|A(\theta_1)|}.\]

\(A\) and \(C\) are defined as in roy_pdf_cdf_sf().

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

dist_roy.qtf(q)#

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

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 by Chiani.

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

dist_roy.isf(q)#

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

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

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

dist_roy.c_x(t)#

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

\[C_X(t) = \int_{0}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", roy_largest_root(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_roy.m_x(t)#

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

\[M_X(t) = \int_{0}^{\infty} e^{tx} \text{pdf}_X(x) \mathrm{d} x\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", roy_largest_root(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_roy.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 Roy’s largest root:

\[K_X(t) = \log\left(M_X(t)\right)\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", roy_largest_root(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_roy.moments(k)#

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

\[\mu'_X(r) = \int_{0}^{\infty} x^r \text{pdf}_X(x) \mathrm{d} x\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", roy_largest_root(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_roy.cumulants(k)#

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

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