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.

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 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 $$\chi^2_m(\lambda)$$. 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 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.

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.