Mann-Whitney U distribution, continuous data

class ctx.dist_mann_whitney_u(N1, N2)

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 \(N_1 \ge 1\) and \(N_2 \ge 1\) and the support interval \((0, N_1 \cdot N_2))\). See also Wikipedia [1277], R (Statistical System) [554], Murakami and Kamakura [442], vandeWiel [860], Robillard [507] and Zimmermann [1657].

Let \(x_1,\ldots,x_{N_1}\) and \(y1,\ldots,y_{N_2}\) be two sets of measurements, which we denote by \(X\) and \(Y\). The test criterion \(U\) of the Mann-Whitney test is then

\[U = \sum_{j=1}^{N_1} \sum_{k=1}^{N_2} \text{sgn}(x_k - y_k)\]

dist_mann_whitney_u.pmf(x)

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

\[\text{pmf}_X(x) = \sum_{j=x}^{N_1 \cdot N_2} (-1)^{x+j} \binom{j}{x} \frac{\mu'_{[j]}}{j!},\]

where \(\mu'_{[j]}\) is the \(j^{\text{th}}\) factorial moment.

The factorial moments are calculated from the cumulants (see factorial_moments_from_cumulants()).

The pmf can also be calculated from the characteristic function \(C_X(t)\):

\[\text{pmf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( e^{-itx} C_X(t) \right) \mathrm{d} t,\]

>>> 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.cdf(x)

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

\[\text{cdf}_X(x) = \sum_{j=x}^{N_1 \cdot N_2} (-1)^{x+j} \binom{j-1}{x-1} \frac{\mu'_{[j]}}{j!},\]

where \(\mu'_{[j]}\) is the \(j^{\text{th}}\) factorial moment.

The cdf can also be calculated from the characteristic function \(C_X(t)\):

\[\text{cdf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( C_X(t) \sum_{z=0}^x e^{-itz} \right) \mathrm{d} t.\]

>>> 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.sf(x)

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

\[\text{sf}_X(x) = 1-\sum_{j=x}^{N_1 \cdot N_2} (-1)^{x+j} \binom{j-1}{x-1} \frac{\mu'_{[j]}}{j!},\]

where \(\mu'_{[j]}\) is the \(j^{\text{th}}\) factorial moment.

The cdf can also be calculated from the characteristic function \(C_X(t)\):

\[\text{sf}(x) = 1-\text{cdf}(x) = \frac{1}{\pi} \int_{0}^{\pi} \Re \left( C_X(t) \sum_{z=x+1}^{N_1 \cdot N_2} e^{-itz} \right) \mathrm{d} t.\]

>>> 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.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a Mann-Whitney \(U\) distribution. There is no closed form for the qtf: It is computed using the Brent algorithm with starting values from a Cornish-Fisher or Jensen approximation.

>>> 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.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a Mann-Whitney \(U\) distribution. There is no closed form for the isf: It is computed using the Brent algorithm with starting values from a Cornish-Fisher or Jensen approximation.

>>> 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.g_x(t)

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

\[G_X(t) = \frac{1}{\binom{N_1+N_2}{N_1}} \frac{\prod_{r=N_1+1}^{N_1+N_2}(1-x^r)}{\prod_{r=1}^{N_2}(1-x^r)}.\]

See also: v.d.Wiel, p. 14, equ. 2.5

>>> 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.c_x(t)

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

\[C_X(t) = \prod_{r=1}^{N_2} \frac{r}{N_2 + r} \frac{1-\exp((N_2+r)it)}{1-\exp(itr)}.\]

>>> 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.m_x(t)

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

\[M_X(t) = \prod_{r=1}^{N_2} \frac{r}{N_2 + r} \frac{1-\exp((N_2+r)t)}{1-\exp(tr)}.\]

>>> 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.k_x(t, k=0)

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

\[K_X(t) = \sum_{r=1}^{N_2} \log \left( \frac{r}{N_2 + r} \frac{1-\exp((N_2+r)t)}{1-\exp(tr)} \right).\]

\(K_X(t)\), 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 Mann-Whitney U distribution, are defined as

\[K_X(t) = \sum_{r=1}^{m} \log \left( \frac{r}{n + r} \frac{1-\exp((n+r)t)}{1-\exp(t \cdot r)} \right),\]

\[K_X^{(1)}(s) = \sum_{r=1}^{m} \left( \frac{n+r}{1-\exp((n+r)s)} - \frac{n(\exp(r \cdot s)+n+r)}{1-\exp(r \cdot s)} \right),\]

See also Murakami1_new in DistCiornishArb.vb.

\[K_X^{(j)}(s) = (-1)^j \sum_{r=1}^{m} \sum_{i=0}^{1} (-1)^i \cdot t_i^j \cdot \sum_{k=1}^{j} c(j-2,k) \cdot z_i^j, \quad j \ge 2, \quad \text{where}\]

\[\begin{split}z_i = \frac{1}{1-\exp(t_i \cdot s)}, \quad t_i = \begin{cases} n+r, & i=0,\\ r & i=1, \end{cases}\end{split}\]

and the coefficients \(c(i,j)\) are calculated recursively, with \(c(0,1) = c(0,2)=c(i,1)=1\), and

\[c(i,j) = (j-1) \cdot c(i-1,j-1) + j \cdot c(i-1,j), \quad j \ge 2.\]

See also Murakami2_deriv2 in DistCiornishArb.vb.

The saddlepoint \(s\) is determined numerically using Newton iterations, with a starting value of \(s=0.1\).

>>> 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.moments(k)

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

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Mann-Whitney \(U\) distribution. The cumulants of \(U\) are given by (Robillard1972):

\begin{eqnarray} \kappa_{2j} & = &\frac{B_{2j}}{2j} \left[ \sum_{s=N_1+1}^{N_1+N_2} s^{2j} - \sum_{s=1}^{N_2} s^{2j} \right] \nonumber \\ & = &\frac{B_{2j}}{2j(2j+1)} \left[ B_{2j+1}(N_2+N_1+1) + B_{2j+1} - B_{2j+1}(N_1+1) - B_{2j+1}(N_2+1) \right] \nonumber \end{eqnarray}

and \(\kappa_{2j+1}=0\), \(j \geq 1\), and \(B_{2j}\) and \(B_{2j}(x)\) are the Bernoulli numbers and polynomials, respectively, of degree \(2j\).

>>> 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

Approximations

ctx.mannwhitney_ft(x, n, results='cdf')

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

Calculates the pdf, cdf and sf from the characteristic function (see pmf_from_cf_lattice() and cdf_from_cf_lattice()).

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

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

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

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

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

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

ctx.mannwhitney_spa(x, m, 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.

The saddlepoint \(s\) is determined numerically using Newton iterations, with a starting value of \(s=0.1\).

ctx.mannwhitney_spa_inv(q, m, n, results='qtf')

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

Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.