Discrete (non-lattice) distribution functions related to rank tests#
Cochran \(S\) distribution (under \(H_0\)), pmf vector#
- math53lib.cochran_s_pmf_vector(x, n, lambda)#
Returns the vector of all pmf values of the Cochran \(S\) 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], vandeWiel [860], and Skillings [527].
Returns \(\text{pmf}_X(x)\), 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 \(p(\cdot)\).
Friedman \(S\) distribution (under \(H_0\)), pmf vector#
- math53lib.friedman_s_pmf_vector(x, n, lambda)#
Returns the vector of all pmf values of the Friedman \(S\) 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: https://www.statsdirect.com/help/nonparametric_methods/friedman.htm
See also Wikipedia [1273], vandeWiel [860] and Skillings [527].
Returns \(\text{pmf}_X(x)\), 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 \(p(\cdot)\).
Quade \(S\) distribution (under \(H_0\)), pmf vector#
- math53lib.quade_s_pmf_vector(x, n, lambda)#
Returns the vector of all pmf values of the Quade \(S\) distribution 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], vandeWiel [860], and Skillings [527] .
Returns \(\text{pmf}_X(x)\), 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 \(p(\cdot)\).