Generalized Page \(L\) distribution, continuous data

class ctx.dist_page_l(k, n)

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

The distribution of Page’s \(L\) is a discrete (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_page_l.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following a generalized Page \(L\) distribution.

\[\text{pmf}_X(x) = ??\]

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

dist_page_l.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a generalized Page \(L\) distribution.

\[\text{cdf}_X(x) = ??\]

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

dist_page_l.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a generalized 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: ", page_l_continuous(mu, sigma).pmf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_page_l.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a generalized 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: ", page_l_continuous(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_page_l.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a generalized 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: ", page_l_continuous(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_page_l.c_x(t)

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

\[C_X(t) = ??\]

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

dist_page_l.m_x(t)

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

\[M_X(t) = ??\]

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

dist_page_l.k_x(t, k=0)

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

\[K_X(t) = ??\]

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

dist_page_l.moments(k)

Returns the first \(j\) moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a generalized 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: ", page_l_continuous(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_page_l.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a generalized 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: ", page_l_continuous(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00

Approximations

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

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

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

For the Page L distribution, the cumulants are calculated from the pmf

ctx.page_ecf_inv(q, f, rho, omega)

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

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

For the Page L distribution, the cumulants are calculated from the pmf

ctx.page_spa(x, n, results='c')

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

Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf.

\(K_X(t)\), the cumulant generating function, and its \(j^{\text{th}}\) derivatives, \(K_X^{(j)}(t), j = 1 \ldots k\), of a random variable \(X\), following a Jonckheere-Terpsta S distribution, are calculated from the pgf. The saddlepoint \(s\) is determined numerically using Newton iterations, with a starting value of \(s=0.1\).

ctx.page_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.