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 = χ m 2 ( λ ) χ m 2 ( λ ) + χ n 2 , {\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}
where χ m 2 ( λ ) {\displaystyle \chi _{m}^{2}(\lambda )} is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter λ {\displaystyle \lambda } , and χ n 2 {\displaystyle \chi _{n}^{2}} is a central chi-squared random variable with degrees of freedom n, independent of χ m 2 ( λ ) {\displaystyle \chi _{m}^{2}(\lambda )} . In this case, X ∼ Beta ( m 2 , n 2 , λ ) {\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
Y = χ n 2 χ n 2 + χ m 2 ( λ ) , {\displaystyle 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 {\displaystyle Y} follows the type II distribution, then X = 1 − Y {\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:
F ( x ) = ∑ j = 0 ∞ P ( j ) I x ( α + j , β ) , {\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 I x ( a , b ) {\displaystyle I_{x}(a,b)} is the incomplete beta function. That is,
F ( x ) = ∑ j = 0 ∞ 1 j ! ( λ 2 ) j e − λ / 2 I x ( α + j , β ) . {\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
F ( x ) = ∑ j = 0 ∞ P ( j ) I x ( α , β + j ) . {\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 and Chattamvelli.
Probability density function The (Type I) probability density function for the noncentral beta distribution is:
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