Discrete (lattice) distribution functions related to (stratified) rank tests

Kendall \(S\) (or tau) distribution (under \(H_0\)), pmf vector

math53lib.KendallTauPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Kendall \(S\) (or tau) distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix. See also Wikipedia [1274], Noether [447], vandeWiel [860].

Returns \(\text{pmf}(x)\), the probability mass function (pmf) of a random variable \(X\), following a Kendall tau distribution.

The null distribution can be calculated as follows: Let \(p_N(t) = \text{Pr}[T_N=t]\). Then the following recurrence relation holds:

\[p_N(t) = p_N(t-1) + [p_{N-1}(t) - p_{N-1}(t-N)] /N,\]

where \(p_N(t) = 0\) for \(t<0\) or \(t>N(N-1)/2\), and \(p_N(0)=1/N!\).

Mann-Whitney \(U\) distribution (under \(H_0\)), pmf vector

math53lib.MannWhitneyUPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Mann-Whitney \(U\) distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1277], Murakami and Kamakura [442], Robillard [507], [vandeWiel2000` and Zimmermann [1657].

Returns \(\text{pmf}(x)\), the probability mass function (pmf) of a random variable \(X\), following a Mann-Whitney U distribution. The null distribution of the MW test can be calculated as follows: Let \(p_{n,m}(u)\) denote the probability that \(U=u\) in samples of size \(n\) and \(m\). Then (see Zimmermann [1657])

\[(m+n) p_{n,m}(u) = n p_{n-1,m}(u-m) + m p_{n,m-1}(u) = n p_{n-1,m}(u) + m p_{n,m-1}(u-n)\]

Jonckheere-Terpsta \(S\) distribution (under \(H_0\)), pmf vector

math53lib.JTerpstaSPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Jonckheere-Terpsta \(S\) distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1273], Murakami and Kamakura [442], Robillard [507], [vandeWiel2000` and Skillings [527].

Returns \(\text{pmf}(x)\), the probability mass function (pmf) of a random variable \(X\), following a Jonckheere-Terpsta S 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):

\[p(n_1,\ldots,n_k; t) = \sum_{x} p(n_1,\ldots,n_k; x) \times p(n_1,\ldots,n_k; t-x)\]

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

Spearman \(\rho\) distribution (under \(H_0\)), pmf vector

math53lib.SpearmanRhoPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Page \(L\) distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1273], [vandeWiel2000` and Skillings [527] .

Returns \(\text{pmf}(x)\), the probability mass function (pmf) of a random variable \(X\), following a Jonckheere-Terpsta S 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):

\[p(n_1,\ldots,n_k; t) = \sum_{x} p(n_1,\ldots,n_k; x) \times p(n_1,\ldots,n_k; t-x)\]

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

Sign test distribution (under \(H_0\)), pmf vector

math53lib.SigntestPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Sign test distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1287], [vandeWiel2000`, Bennett [34] and Zimmermann [1656].

Returns \(\text{pmf}(x)\), 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 (see Zimmermann [1656]) :

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

Wilcoxon \(T\) distribution (under \(H_0\)), pmf vector

math53lib.WilcoxonPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Wilcoxon distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1287], Fellingham and Stoker [312], [vandeWiel2000`, Bennett [34] and Zimmermann [1656].

Returns \(\text{pmf}(x)\), 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 (see Zimmermann [1656]) :

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

Page \(L\) distribution (under \(H_0\)), pmf vector

math53lib.PageLPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Page \(L\) distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1273], [vandeWiel2000` and Skillings [527].

Returns \(\text{pmf}(x)\), the probability mass function (pmf) of a random variable \(X\), following a Jonckheere-Terpsta S 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 \(n_1,\ldots,n_k\):

\[p(n_1,\ldots,n_k; t) = \sum_{x} p(n_1,\ldots,n_k; x) \times p(n_1,\ldots,n_k; t-x)\]

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

Quade \(L\) distribution (under \(H_0\)), pmf vector

math53lib.QuadeLPmfVector(x, n, lambda)

Returns the vector of all pmf values of the Quade \(L\) distribution under \(H_0\). The vector is returned as a nested list of Decimals or as a mp.matrix or as a iv.matrix.

See also Wikipedia [1273], [vandeWiel2000` and Skillings [527].

Returns \(\text{pmf}(x)\), the probability mass function (pmf) of a random variable \(X\), following a Jonckheere-Terpsta S 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 \(n_1,\ldots,n_k\):

\[p(n_1,\ldots,n_k; t) = \sum_{x} p(n_1,\ldots,n_k; x) \times p(n_1,\ldots,n_k; t-x)\]

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