Noncentral Wilcoxon signed rank T distribution, Bennett alternatives

class ctx.dist_bennett(N)

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

The distribution of Wilcoxon’s signed rank test is a discrete (lattice) probability distribution with sample size \(N > 2\) and the support interval \((0, N(N+1))\). See also Wikipedia [1287], R (Statistical System) [555], Fellingham and Stoker [312], vandeWiel [860], Bennett [34] and Zimmermann [1656].

We consider \(N\) continuously distributed random variables \(D_i,i=1\ldots N\), with common pdf \(h_0\). In a sample \((d_1,\ldots,d_N)\) of size \(N\) let \(r_i\) be the rank of \(d_i\) in the ordered sample.

The test criterion of Wilcoxon’s Signed Rank is \(T_N=\sum_{i=1}^N S(d_i)r_i\), where \(S(d_i)=1\) for \(x>0\) and \(S(d_i)=0\) for \(x<0\). \(T_N\) can assume values between 0 and \(\tfrac{1}{2}N(N+1)\) in steps of 1.

dist_bennett.pmf(x)

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following a Wilcoxon signed rank T distribution. The null distribution can be calculated as follows: Let \(p_N(w)\) denote the probability \(\text{Pr}[W_N=w]\) in a sample of size \(N\). Then the following recurrence relation holds (Zimmermann_1985_dependent) :

\[p_N(w) = \tfrac{1}{2} \left( p_{N-1}(w) + p_{N-1}(w-N)\right).\]

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

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

dist_bennett.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a Wilcoxon signed rank T distribution.

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

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

dist_bennett.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a Wilcoxon signed rank T distribution:

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

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

dist_bennett.qtf(q)

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

dist_bennett.isf(q)

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

dist_bennett.g_x(t)

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following a Wilcoxon signed rank T distribution:

\[G_X(t) = \prod_{h=1}^{n} \left( p t^h + (1-p) \right).\]

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

dist_bennett.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a Wilcoxon signed rank T distribution:

\[C_X(t) = \exp\left( \tfrac{1}{4} n (n+1)it \right) \prod_{h=1}^{n} \cosh\left(\tfrac{h}{2} it\right).\]

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

dist_bennett.m_x(t)

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following a Wilcoxon signed rank T distribution:

\[M_X(t) = (4p(1-p))^{\tfrac{1}{2} n} \exp\left( \tfrac{1}{4} n (n+1)t \right) \prod_{h=1}^{n} \cosh\left(\tfrac{1}{2} (\log(p/(1-p)) + h \cdot t)\right), \quad a =\]

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

dist_bennett.k_x(t, k=0)

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following a Wilcoxon signed rank T distribution:

\[K_X(t) = \tfrac{1}{2} n \log(4p(1-p)) + \tfrac{1}{4} n (n+1)t + \sum_{h=1}^{n} \log \left(\cosh\left(\tfrac{1}{2} (\log(p/(1-p)) + h \cdot t ) \right)\right).\]

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

dist_bennett.moments(k)

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

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

dist_bennett.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a Wilcoxon signed rank T distribution. The cumulants of \(W\) are given by (Fellingham_1964) :

Central case:

\[\kappa_{2j}(W_N) = \frac{2^{2j} (2^{2j}-1) B_{2j}}{2j} \sum_{i=1}^N r_i^{2j} = \frac{2^{2j} (2^{2j}-1) B_{2j}}{2j} \frac{B_{2j+1}(N+1)-B_{2j+1}}{2j+1}, \quad \text{and}\]

\[\kappa_{2j+1}(W_N) = 0, \quad \text{for } j \geq 0.\]

Noncentral case:

\[\kappa_1 = \tfrac{1}{4} n(n+1) (1+\tanh(0.5 a) ) = \tfrac{1}{2} n(n+1) p\]

\[\kappa_2 = \tfrac{1}{6} n(n+1)(2n+1) p(1-p)\]

\[\kappa_3 = \tfrac{1}{4} n^2 (n+1)^2 pq(q-p)\]

\[\kappa_4 = \tfrac{1}{30} n(n+1)(2n+1)(3n^2+3n-1)pq(1-6pq)\]

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