Noncentral hypergeometric distribution, Fisher alternatives

class ctx.dist_hypergeo_nc_fisher(n1, m1, N, theta)

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

The noncentral hypergeometric distribution (Fisher alternatives) is the conditional distribution of one of two binomial random variables \(X_1\) and \(X_2\), given that their sum is fixed. The distribution arises as the power function for Fisher’s (1934) exact test of independence in a \(2 \times 2\) contingency table.

If \(X_i\) has parameters \(n_i\), \(p_i = 1 - q_i\), and \(N = n_1 + n_2\), \(\theta = p_1 q_2 /(q_1p_2)\) then

\[Pr[X_1 = x|X_1 + X_2 = m_1] = \text{h}(x; n_1, m_1, N, \theta),\]

where \(\text{h}(x; n_1, m_1, N, \theta)\) is given below. There are other noncentral hypergeometric distributions as well.

See also: Wikipedia [1316], Johnson et al. [411] page 293.

dist_hypergeo_nc_fisher.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[\text{pmf}_X(x) = \text{h}(x; n_1, m_1, N, \theta) = \frac{\binom{n_1}{x} \binom{n_2}{m_1-x}\theta^x}{\binom{n_2}{m_1} {}_2F_1(-n_1, -m_1;n_2+1-m_1; \theta)},\]

where \(\theta = p_1 q_2 /(q_1p_2)\) and \(\text{max}(0, m_1-n_2 )\le x \le \text{min}(n_1,m_1)\).

The following recursions are used for the PMF (see Wikipedia):

\[\text{h}(x; n_1, m_1, N, \theta)= \frac{(m_1-x+1)(n_1-x+1) \theta}{x(m_2-n_1+x)} \text{h}(x-1; n_1, m_1, N, \theta)\]

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

dist_hypergeo_nc_fisher.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[\text{cdf}_X(x) = \sum_{j=\max(0,n+K-N)}^{k} \text{pmf}_X(j).\]

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

dist_hypergeo_nc_fisher.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[\text{sf}_X(x) = \sum_{j=k+1}^{\min(K,n)} \text{pmf}_X(j).\]

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

dist_hypergeo_nc_fisher.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives).

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

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

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.g_x(t)

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[G(t) = \frac{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta t)}{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[G(z) = \frac{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta z)}{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta)}\]

\[C_X(t) = G(e^{it}) .\]

\[C_X(t) = \frac{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta e^{it})}{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[M_X(t) = G(e^{t}).\]

\[M_X(t) = \frac{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta e^{t})}{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives):

\[K_X(t) = \log \left[ G(e^{t}) \right].\]

\[K_X(t) = \log \left[ \frac{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta e^{t})}{{}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta)} \right].\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives) (Wikipedia). The raw moments are calculated from the factorial moments:

\[\mu'_{[r]} = \frac{n_1! m_1! (n_2-m_1)! \cdot {}_2F_1(r-n_1, r-m_1; r+n_2+1-m_1; \theta)}{(n_1-r)! (m_1-r)! (n_2-m_1+r)! \cdot {}_2F_1(-n_1, -m_1; n_2+1-m_1; \theta)}\]

where \({}_2F_1(\cdot)\) is the Gauss hypergeometric function (see hyp2f1().)

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

dist_hypergeo_nc_fisher.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an noncentral hypergeometric distribution (Fisher alternatives). The cumulants are calculated from the moments.

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