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:3
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 Posten4 and Chattamvelli.5
Probability density function
The (Type I) probability density function for the noncentral beta distribution is:
f ( x ) = ∑ j = 0 ∞ 1 j ! ( λ 2 ) j e − λ / 2 x α + j − 1 ( 1 − x ) β − 1 B ( α + j , β ) . {\displaystyle 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 {\displaystyle B} is the beta function, α {\displaystyle \alpha } and β {\displaystyle \beta } are the shape parameters, and λ {\displaystyle \lambda } is the noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.6
Related distributions
Transformations
If X ∼ Beta ( α , β , λ ) {\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)} , then β X α ( 1 − X ) {\displaystyle {\frac {\beta X}{\alpha (1-X)}}} follows a noncentral F-distribution with 2 α , 2 β {\displaystyle 2\alpha ,2\beta } degrees of freedom, and non-centrality parameter λ {\displaystyle \lambda } .
If X {\displaystyle X} follows a noncentral F-distribution F μ 1 , μ 2 ( λ ) {\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)} with μ 1 {\displaystyle \mu _{1}} numerator degrees of freedom and μ 2 {\displaystyle \mu _{2}} denominator degrees of freedom, then
Z = μ 2 μ 1 μ 2 μ 1 + X − 1 {\displaystyle Z={\cfrac {\cfrac {\mu _{2}}{\mu _{1}}}{{\cfrac {\mu _{2}}{\mu _{1}}}+X^{-1}}}}follows a noncentral Beta distribution:
Z ∼ Beta ( 1 2 μ 1 , 1 2 μ 2 , λ ) {\displaystyle 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 λ = 0 {\displaystyle \lambda =0} , the noncentral beta distribution is equivalent to the (central) beta distribution.
Citations
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."
References
Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. /wiki/Doi_(identifier) ↩
Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. /wiki/Doi_(identifier) ↩
Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. /wiki/Doi_(identifier) ↩
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. /wiki/Doi_(identifier) ↩
Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. /wiki/Doi_(identifier) ↩
Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151. /wiki/Doi_(identifier) ↩