Cochran-Friedman-Quade distribution

class ctx.dist_friedman(k, n)

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

The distribution of the Cochran-Friedman-Quade tests is a discrete (non-lattice) probability distribution with k samples of size \(n_1 \ge 1, \ldots, n_k \ge 1\) and the support interval \((0, n m))\). See also Wikipedia [1273], Noether [447], vandeWiel [860] .

Consider \(k\) independent groups \(X_i\) of sizes \(n_i, i=1 \ldots k\). The Page \(L\) statistic is defined as

\[L = ??\]

dist_friedman.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following a Page \(L\) distribution. Let \(p(n_1,\ldots,n_k; t) = \text{Pr}[J_N=t]\). If \(J_N\) is based on \(k\) independent samples of sizes \(n_1,\ldots,n_k\), then (Skillings 1980):

\[p(n_1,\ldots,n_k; t) = ??\]

where the sum is over all \(x\) with positive \(p(\cdot)\).

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

dist_friedman.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Page \(L\) distribution. Let \(p(n_1,\ldots,n_k; t) = \text{Pr}[J_N=t]\). If \(J_N\) is based on \(k\) independent samples of sizes \(n_1,\ldots,n_k\), then (Skillings 1980):

\[p(n_1,\ldots,n_k; t) = ??\]

where the sum is over all \(x\) with positive \(p(\cdot)\).

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

dist_friedman.sf(x)

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

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x)\]

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

dist_friedman.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Page \(L\) distribution. There is no closed form for the qtf: It is computed with Newton iterations where the starting values are from Boost.

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

dist_friedman.isf(q)

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

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

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

dist_friedman.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Page \(L\) distribution:

\[C_X(t) = ??\]

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

dist_friedman.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a Page \(L\) distribution:

\[M_X(t) = ??\]

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

dist_friedman.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a Page \(L\) distribution:

\[K_X(t) = ??\]

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

dist_friedman.moments(k)

Returns the first \(j\) moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Page \(L\) distribution (Wikipedia). The moments are calculated from the cumulants.

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

dist_friedman.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Page \(L\) distribution. The cumulants of \(J_N\) are given by :

\[\kappa_{2j} = ??\]

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