Noncentral beta distribution

{{Probability distribution |

name =Noncentral Beta|

type = density|

notation = Beta(α, β, λ)|

parameters = α > 0 shape (real)
β > 0 shape (real)
λ ≥ 0 noncentrality (real)|

support = $x \in [0; 1]\!$ |

pdf = (type I) $\sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!}\frac{x^{\alpha + j - 1}\left(1-x\right)^{\beta - 1}}{\mathrm{B}\left(\alpha + j,\beta\right)}$ |

cdf = (type I) $\sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!} I_x \left(\alpha + j,\beta\right)$ |

mean = (type I) $e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha + 1\right)}{\Gamma\left(\alpha\right)} \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha + \beta + 1\right)} {}_2F_2\left(\alpha+\beta,\alpha+1;\alpha,\alpha+\beta+1;\frac{\lambda}{2}\right)$ (see Confluent hypergeometric function)|

variance = (type I) $e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha + 2\right)}{\Gamma\left(\alpha\right)} \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha + \beta + 2\right)} {}_2F_2\left(\alpha+\beta,\alpha+2;\alpha,\alpha+\beta+2;\frac{\lambda}{2}\right) - \mu^2$ where $\mu$ is the mean. (see Confluent hypergeometric function)

}}

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

: $X = \frac{\chi^2_m(\lambda)}{\chi^2_m(\lambda) + \chi^2_n},$

where $\chi^2_m(\lambda)$ is a [[Noncentral chi-squared distribution|

noncentral chi-squared]] random variable with degrees of freedom m and noncentrality parameter $\lambda$ , and $\chi^2_n$ is a central chi-squared random variable with degrees of freedom n, independent of $\chi^2_m(\lambda)$ .{{cite journal

|first=R.

|last=Chattamvelli

|title=A Note on the Noncentral Beta Distribution Function

|journal=The American Statistician

|volume=49

|issue= 2

|year=1995

|pages= 231–234

|doi=10.1080/00031305.1995.10476151}}

In this case, $X \sim \mbox{Beta}\left(\frac{m}{2},\frac{n}{2},\lambda\right)$

A Type II noncentral beta distribution is the distribution

of the ratio

: $Y = \frac{\chi^2_n}{\chi^2_n + \chi^2_m(\lambda)},$

where the noncentral chi-squared variable is in the denominator only. 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:

: $F(x) = \sum_{j=0}^\infty P(j) I_x(\alpha+j,\beta),$

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,

: $F(x) = \sum_{j=0}^\infty \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}I_x(\alpha+j,\beta).$

The Type II cumulative distribution function in mixture form is

: $F(x) = \sum_{j=0}^\infty P(j) I_x(\alpha,\beta+j).$

Algorithms for evaluating the noncentral beta distribution functions are given by Posten{{cite journal|first=H.O.|last=Posten|title=An Effective Algorithm for the Noncentral Beta Distribution Function|journal=The American Statistician|year= 1993|volume= 47|issue= 2|pages=129–131|jstor=2685195|doi=10.1080/00031305.1993.10475957}} and Chattamvelli.

Probability density function

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

: $f(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}\frac{x^{\alpha+j-1}(1-x)^{\beta-1}}{B(\alpha+j,\beta)}.$

where $B$ is the beta function, $\alpha$ and $\beta$ are the shape parameters, and $\lambda$ is the noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.

Related distributions

= Transformations =

If $X\sim\mbox{Beta}\left(\alpha,\beta,\lambda\right)$ , then $\frac{\beta X}{\alpha (1-X)}$ follows a noncentral F-distribution with $2\alpha, 2\beta$ degrees of freedom, and non-centrality parameter $\lambda$ .

If $X$ follows a noncentral F-distribution $F_{\mu_{1}, \mu_{2}}\left( \lambda \right)$ with $\mu_{1}$ numerator degrees of freedom and $\mu_{2}$ denominator degrees of freedom, then

: $Z = \cfrac{\cfrac{\mu_{2}}{\mu_{1}}}{\cfrac{\mu_{2}}{\mu_{1}} + X^{-1} }$

follows a noncentral Beta distribution:

: $Z \sim \mbox{Beta}\left(\frac{1}{2}\mu_{1},\frac{1}{2}\mu_{2},\lambda\right)$ .

This is derived from making a straightforward transformation.

= Special cases =

When $\lambda = 0$ , the noncentral beta distribution is equivalent to the (central) beta distribution.

References

= Citations =

= Sources =

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

Category:Continuous distributions