Evaluation of the Noncentral t Distribution with S‐Systems

Densities, cumulatives, and quantiles of the noncentral distributions are notoriously difficult to calculate. It is shown how the noncentral t distribution is represented in the canonical S‐system form, i.e., as a set of first‐order nonlinear differential equations in which the derivative of each variable equals a difference of two products of power‐law functions. By solving the S‐system numerically one rapidly obtains densities, cumulatives, quantiles, and moments of the central and noncentral t distributions, as well as power functions for the t‐test.