Noncentral beta distribution

Noncentral Beta
Notation	Beta(α, β, λ)
Parameters	α > 0 shape (real); β > 0 shape (real); λ ≥ 0 noncentrality (real)
Support
PDF	(type I)
CDF	(type I)
Mean	(type I) (see Confluent hypergeometric function)
Variance	(type I) where 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 $\chi _{m}^{2}(\lambda )$ .[1] 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.[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 _{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[2] and Chattamvelli.[1]

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.[1]

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

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

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."

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[Chat-1] Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151.

[2] Posten, H.O. (1993). "An Effective Algorithm for the Noncentral Beta Distribution Function". The American Statistician. 47 (2): 129–131. doi:10.1080/00031305.1993.10475957. JSTOR 2685195.

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)