Noncentral beta distribution

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

Noncentral Beta

Notation	Beta(α, β, λ)
Parameters	α > 0 shape (real) β > 0 shape (real) λ ≥ 0 noncentrality (real)
Support	${\displaystyle x\in [0;1]\!}$
PDF	(type I) ${\displaystyle \sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}{\frac {x^{\alpha +j-1}\left(1-x\right)^{\beta -1}}{\mathrm {B} \left(\alpha +j,\beta \right)}}}$
CDF	(type I) ${\displaystyle \sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}I_{x}\left(\alpha +j,\beta \right)}$
Mean	(type I) ${\displaystyle e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +1\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +1\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +1;\alpha ,\alpha +\beta +1;{\frac {\lambda }{2}}\right)}$ (see Confluent hypergeometric function)
Variance	(type I) ${\displaystyle e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +2\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +2\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +2;\alpha ,\alpha +\beta +2;{\frac {\lambda }{2}}\right)-\mu ^{2}}$ where ${\displaystyle \mu }$ is the mean. (see Confluent hypergeometric function)

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

${\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}$

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

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

${\displaystyle Y={\frac {\chi _{n}^{2}}{\chi _{n}^{2}+\chi _{m}^{2}(\lambda )}},}$

where the noncentral chi-squared variable is in the denominator only.^[1] If ${\displaystyle Y}$ follows the type II distribution, then ${\displaystyle X=1-Y}$ 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]

${\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha +j,\beta ),}$

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 ${\displaystyle I_{x}(a,b)}$ is the incomplete beta function. That is,

${\displaystyle F(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}I_{x}(\alpha +j,\beta ).}$

The Type II cumulative distribution function in mixture form is

${\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha ,\beta +j).}$

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

Probability density function

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

${\displaystyle f(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}{\frac {x^{\alpha +j-1}(1-x)^{\beta -1}}{B(\alpha +j,\beta )}}.}$

where ${\displaystyle B}$ is the beta function, ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are the shape parameters, and ${\displaystyle \lambda }$ 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 ${\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)}$, then ${\displaystyle {\frac {\beta X}{\alpha (1-X)}}}$ follows a noncentral F-distribution with ${\displaystyle 2\alpha ,2\beta }$ degrees of freedom, and non-centrality parameter ${\displaystyle \lambda }$.

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

${\displaystyle Z={\cfrac {\cfrac {\mu _{2}}{\mu _{1}}}{{\cfrac {\mu _{2}}{\mu _{1}}}+X^{-1}}}}$

follows a noncentral Beta distribution:

${\displaystyle Z\sim {\mbox{Beta}}\left({\frac {1}{2}}\mu _{1},{\frac {1}{2}}\mu _{2},\lambda \right)}$.

This is derived from making a straightforward transformation.

Special cases

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