Noncentral distribution

Noncentral distributions are families of probability distributions that are related to other "central" families of distributions by means of a noncentrality parameter. Whereas the central distribution describes how a test statistic is distributed when the difference tested is null, noncentral distributions describe the distribution of a test statistic when the null is false (so the alternative hypothesis is true). This leads to their use in calculating statistical power. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family. For example, the Student's t-distribution is the central family of distributions for the noncentral t-distribution family. Noncentrality parameters are used in the following distributions: Noncentral t-distribution Noncentral chi-squared distribution Noncentral chi-distribution Noncentral F-distribution Noncentral beta distribution In general, noncentrality parameters occur in distributions that are transformations of a normal distribution. The "central" versions are derived from normal distributions that have a mean of zero; the noncentral versions generalize to arbitrary means. For example, the standard (central) chi-squared distribution is the distribution of a sum of squared independent standard normal distributions, i.e., normal distributions with mean 0, variance 1. The noncentral chi-squared distribution generalizes this to normal distributions with arbitrary mean and variance. Each of these distributions has a single noncentrality parameter. However, there are extended versions of these distributions which have two noncentrality parameters: the doubly noncentral beta distribution, the doubly noncentral F distribution and the doubly noncentral t distribution. These types of distributions occur for distributions that are defined as the quotient of two independent distributions. When both source distributions are central (either with a zero mean or a zero noncentrality parameter, depending on the type of distribution), the result is a central distribution.