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|newpage| Discrete (lattice) distribution functions related to (stratified) rank tests ======================================================================================== Kendall `S` (or \tau) distribution (under `H_0`), pmf vector ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis27`, :cite:t:`Noether1967`, :cite:t:`vandeWiel2000`. 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: .. math:: 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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis28`, :cite:t:`Murakami2009`, :cite:t:`Robillard1972`, [vandeWiel2000` and :cite:t:`Zimmermann1985b`. 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 :cite:t:`Zimmermann1985b`) .. math:: (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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis29`, :cite:t:`Murakami2009`, :cite:t:`Robillard1972`, [vandeWiel2000` and :cite:t:`Skillings1980`. 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): .. math:: 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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis29`, [vandeWiel2000` and :cite:t:`Skillings1980` . 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): .. math:: 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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis26`, [vandeWiel2000`, :cite:t:`Bennett1972` and :cite:t:`Zimmermann1985a`. 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 :cite:t:`Zimmermann1985a`) : .. math:: 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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis26`, :cite:t:`Fellingham1964`, [vandeWiel2000`, :cite:t:`Bennett1972` and :cite:t:`Zimmermann1985a`. 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 :cite:t:`Zimmermann1985a`) : .. math:: 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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis29`, [vandeWiel2000` and :cite:t:`Skillings1980`. 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`: .. math:: 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 ------------------------------------------------------------------------------- .. method:: 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 :cite:p:`WikipediaDis29`, [vandeWiel2000` and :cite:t:`Skillings1980`. 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`: .. math:: 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)`.