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) $\sum _{j=0}^{\mathrm{\infty }}{e}^{-\lambda /2}\frac{{\left(\frac{\lambda }{2}\right)}^j}{j!}\frac{x^{\alpha +j-1}{(1-x)}^{\beta -1}}{\mathrm{B}(\alpha +j,\beta )}$
CDF	(type I) $\sum _{j=0}^{\mathrm{\infty }}{e}^{-\lambda /2}\frac{{\left(\frac{\lambda }{2}\right)}^j}{j!}{I}_x(\alpha +j,\beta )$
Mean	(type I) $e^{-\frac{\lambda }{2}}\frac{\mathrm{\Gamma }(\alpha +1)}{\mathrm{\Gamma }\left(\alpha \right)}\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha +\beta +1)}{}_2{F}_2(\alpha +\beta ,\alpha +1;\alpha ,\alpha +\beta +1;\frac{\lambda }{2})$ (see Confluent hypergeometric function)
Variance	(type I) $e^{-\frac{\lambda }{2}}\frac{\mathrm{\Gamma }(\alpha +2)}{\mathrm{\Gamma }\left(\alpha \right)}\frac{\mathrm{\Gamma }(\alpha +\beta )}{\mathrm{\Gamma }(\alpha +\beta +2)}{}_2{F}_2(\alpha +\beta ,\alpha +2;\alpha ,\alpha +\beta +2;\frac{\lambda }{2})-{\mu }^2$ 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, $X\sim \text{Beta}(\frac{m}{2},\frac{n}{2},\lambda )$

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:

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

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,

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

The Type II cumulative distribution function in mixture form is

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

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:

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

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 $X\sim \text{Beta}(\alpha ,\beta ,\lambda )$, 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

$Z=\frac{\frac{{\mu }_2}{{\mu }_1}}{\frac{{\mu }_2}{{\mu }_1}+X^{-1}}$

follows a noncentral Beta distribution:

$Z\sim \text{Beta}(\frac{1}{2}{\mu }_1,\frac{1}{2}{\mu }_2,\lambda )$.

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."