Noncentral beta distribution explained

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

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

X = (\chi ^{2} _{m( λ )})/(\chi ^{2} _{m( λ )} + \chi ^{2} _{n}) ,

where

\chi ^{2} _{m( λ )} is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter λ , and \chi ^{2} _{n} is a central chi-squared random variable with degrees of freedom n, independent of \chi ^{2} _{m( λ )} .^[1] In this case, X \sim Beta \left( (m)/(, 2) (n)/(2) , λ \right)

A Type II noncentral beta distribution is the distributionof the ratio

Y = (\chi ^{2} _{n})/(\chi ^{2} _{n} + \chi ^{2} _{m( λ )}) , where the noncentral chi-squared variable is in the denominator only.^[1] If Y follows the type II distribution, then X = 1 - Y follows a type I distribution.

Cumulative distribution function

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

F(x) = \sum ^{inf ty} _{j =0} P(j) I _{x(\alpha+j,\beta),} 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 I _{x(a,b)} is the incomplete beta function. That is,F(x) = \sum ^{inf ty} _{j =0} (1)/(\left( j!) (λ)/(2) \right) ^{je} ^{-λ/2} I _{x(\alpha+j,\beta).}

The Type II cumulative distribution function in mixture form is

F(x) = \sum ^{inf ty} _{j =0} P(j) I _{x(\alpha,\beta+j).}

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

Probability density function

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

f(x) = \sum ^{inf in} _{j =0} (1)/(\left( j!) (λ)/(2) \right) ^{je} ^{-λ/2} (x ^{\alpha+j-1} ( 1- x) ^{\beta-1})/(B(\alpha + j,\beta)) .

where

B is the beta function, \alpha and \beta are the shape parameters, and λ 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

Transformations

If

X\sim Beta \left(\alpha,\beta, λ \right) , then (\beta X)/(\alpha ( 1- X)) follows a noncentral F-distribution with 2 \alpha, 2 \beta degrees of freedom, and non-centrality parameter λ .

If

X follows a noncentral F-distribution F _{\mu _{1} , \mu _{2}} \left( λ \right) with \mu _{1} numerator degrees of freedom and \mu _{2} denominator degrees of freedom, thenZ = \cfrac{\cfrac{\mu _{2} }} follows a noncentral Beta distribution:Z \sim Beta \left( (1)/(2) \mu _{1} , (1)/(2) \mu _{2} , λ \right) .This is derived from making a straightforward transformation.

Special cases

When

λ = 0 , the noncentral beta distribution is equivalent to the (central) beta distribution.

References

Sources

• M. Abramowitz and I. Stegun, editors (1965) "Handbook of Mathematical Functions", Dover: New York, NY.
• J.L. Jr . Hodges . On the noncentral beta-distribution . Annals of Mathematical Statistics . 1955 . 26 . 4 . 648–653 . 10.1214/aoms/1177728424 . free .
• G.A.F. . Seber . The non-central chi-squared and beta distributions . . 1963 . 50 . 3–4 . 542–544 . 10.1093/biomet/50.3-4.542 .
• Christian Walck, "Hand-book on Statistical Distributions for experimentalists."

Notes and References

1. R.. Chattamvelli. A Note on the Noncentral Beta Distribution Function. The American Statistician. 49. 2. 1995. 231–234. 10.1080/00031305.1995.10476151.
2. H.O.. Posten. An Effective Algorithm for the Noncentral Beta Distribution Function. The American Statistician. 1993. 47. 2. 129–131. 2685195. 10.1080/00031305.1993.10475957.