Noncentral beta distribution

Noncentral Beta

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) $e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +1\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +1\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +1;\alpha ,\alpha +\beta +1;{\frac {\lambda }{2}}\right)$ (see Confluent hypergeometric function)
Variance	(type I) $e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +2\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +2\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +2;\alpha ,\alpha +\beta +2;{\frac {\lambda }{2}}\right)-\mu ^{2}$ 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

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

where $\chi _{m}^{2}(\lambda )$ is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter $\lambda$, and $\chi _{n}^{2}$ is a central chi-squared random variable with degrees of freedom n, independent of $\chi _{m}^{2}(\lambda )$. 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

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

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,

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

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 and Chattamvelli.

Probability density function

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

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

where $B$ 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

Sources

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