 Noncentrality parameter

Noncentrality parameters are parameters of families of probability distributions which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family.^{[1]} For example, the Student's tdistribution is the central family of distributions for the Noncentral tdistribution family.
Noncentrality parameters are used in the following distributions:

 Noncentral tdistribution
 Noncentral chisquared distribution
 Noncentral chidistribution
 Noncentral Fdistribution
 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) chisquared distribution is the distribution of a sum of squared independent standard normal distributions, i.e. normal distributions with mean 0, variance 1. The noncentral chisquared 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.^{[2]} 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. When one is noncentral, a (singly) noncentral distribution results, while if both are noncentral, the result is a doubly noncentral distribution. As an example, a tdistribution is defined (ignoring constant values) as the quotient of a normal distribution and the square root of an independent chisquared distribution. Extending this definition to encompass a normal distribution with arbitrary mean produces a noncentral tdistribution, while further extending it to allow a noncentral chisquared distribution in the denominator while produces a doubly noncentral tdistribution.
Note also that there are some "noncentral distributions" that are not usually formulated in terms of a "noncentrality parameter": see noncentral hypergeometric distributions, for example.
The noncentrality parameter of the tdistribution may be negative or positive while the noncentral parameters of the other three distributions must be greater than zero.
References
 ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0199206139
 ^ Johnson, N.L., Kotz, S., Balakrishnan N. (1995) Continuous univariate distribitions, Volume 2 (2nd Edition). Wiley. ISBN 0471584940
Categories: Statistical terminology

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