Noncentral Mann-Whitney U distribution, Lehmann alternatives

class ctx.dist_mann_whitney_u_lehmann(m, n)

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

The distribution of Mann-Whitney’s \(U\) is a discrete (lattice) probability distribution with sample sizes \(m \ge 1\) and \(n \ge 1\) and the support interval \((0, n m))\). See also Wikipedia [1277], R (Statistical System) [554], Murakami and Kamakura [442], Robillard [507], vandeWiel [860] and Zimmermann [1657].

Let \(x1,\ldots,x_m\) and \(y1,\ldots,y_n\) be two sets of measurements, which we denote by \(X\) and \(Y\). The test criterion \(U\) of the Mann-Whitney test is then

\[U = \sum_{i=1}^m \sum_{j=1}^n \text{sgn}(x_i - y_j)\]

dist_mann_whitney_u_lehmann.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives.

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

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

dist_mann_whitney_u_lehmann.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives.

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

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

dist_mann_whitney_u_lehmann.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives:

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

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

dist_mann_whitney_u_lehmann.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives. 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: ", mann_whitney_u_continuous(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_mann_whitney_u_lehmann.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives:

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

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

dist_mann_whitney_u_normal.g_x(t)

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives:

\[G_X(t) = ??\]

dist_mann_whitney_u_lehmann.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives:

\[C_X(t) = ??\]

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

dist_mann_whitney_u_lehmann.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives:

\[M_X(t) = ??\]

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

dist_mann_whitney_u_lehmann.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives:

\[K_X(t) = ??\]

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

dist_mann_whitney_u_lehmann.moments(k)

Returns the first \(j\) moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives (Wikipedia). The moments are calculated from the cumulants.

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

dist_mann_whitney_u_lehmann.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Mann-Whitney U distribution under Lehmann alternatives.

kappa_{j} = ??

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