Noncentral beta distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Noncentral Beta
Notation Beta(α, β, λ)
Parameters α > 0 shape (real)
β > 0 shape (real)
λ >= 0 noncentrality (real)
Support
PDF (type I)
CDF (type I)
Mean (type I) (see Confluent hypergeometric function)
Variance (type I) where is the mean. (see Confluent hypergeometric function)

In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.

The noncentral beta distribution (Type I) is the distribution of the ratio

where is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter , and is a central chi-squared random variable with degrees of freedom n, independent of .[1] In this case,

A Type II noncentral beta distribution is the distribution of the ratio

where the noncentral chi-squared variable is in the denominator only.[1] If follows the type II distribution, then follows a type I distribution.

Cumulative distribution function[edit]

The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables:[1]

where λ is the noncentrality parameter, P(.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and is the incomplete beta function. That is,

The Type II cumulative distribution function in mixture form is

Algorithms for evaluating the noncentral beta distribution functions are given by Posten[2] and Chattamvelli.[1]

Probability density function[edit]

The (Type I) probability density function for the noncentral beta distribution is:

where is the beta function, and are the shape parameters, and is the noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.[1]

Related distributions[edit]

Transformations[edit]

If , then follows a noncentral F-distribution with degrees of freedom, and non-centrality parameter .

If follows a noncentral F-distribution with numerator degrees of freedom and denominator degrees of freedom, then follows a noncentral Beta distribution so . This is derived from making a straightforward transformation.

Special cases[edit]

When , the noncentral beta distribution is equivalent to the (central) beta distribution.

References[edit]

  1. ^ a b c d e Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. 
  2. ^ Posten, H.O. (1993). "An Effective Algorithm for the Noncentral Beta Distribution Function". The American Statistician. 47 (2): 129–131. doi:10.1080/00031305.1993.10475957. JSTOR 2685195.