Noncentral beta distribution

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Noncentral Beta
Notation Beta(α, β, λ)
Parameters α > 0 shape (real)
β > 0 shape (real)
λ ≥ 0 noncentrality (real)
Support x ∈ [ 0 ; 1 ] {\displaystyle x\in [0;1]\!} {\displaystyle x\in [0;1]\!}
PDF (type I) ∑ j = 0 ∞ e − λ / 2 ( λ 2 ) j j ! x α + j − 1 ( 1 − x ) β − 1 B ( α + j , β ) {\displaystyle \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)}}} {\displaystyle \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) ∑ j = 0 ∞ e − λ / 2 ( λ 2 ) j j ! I x ( α + j , β ) {\displaystyle \sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}I_{x}\left(\alpha +j,\beta \right)} {\displaystyle \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 − λ 2 Γ ( α + 1 ) Γ ( α ) Γ ( α + β ) Γ ( α + β + 1 ) 2 F 2 ( α + β , α + 1 ; α , α + β + 1 ; λ 2 ) {\displaystyle 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)} {\displaystyle 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 − λ 2 Γ ( α + 2 ) Γ ( α ) Γ ( α + β ) Γ ( α + β + 2 ) 2 F 2 ( α + β , α + 2 ; α , α + β + 2 ; λ 2 ) − μ 2 {\displaystyle 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}} {\displaystyle 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 μ {\displaystyle \mu } {\displaystyle \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 = 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}}},} {\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}

where χ m 2 ( λ ) {\displaystyle \chi _{m}^{2}(\lambda )} {\displaystyle \chi _{m}^{2}(\lambda )} is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter λ {\displaystyle \lambda } {\displaystyle \lambda }, and χ n 2 {\displaystyle \chi _{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 )} {\displaystyle \chi _{m}^{2}(\lambda )}.[1] In this case, X ∼ Beta ( m 2 , n 2 , λ ) {\displaystyle X\sim {\mbox{Beta}}\left({\frac {m}{2}},{\frac {n}{2}},\lambda \right)} {\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 )}},} {\displaystyle 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 {\displaystyle Y} {\displaystyle Y} follows the type II distribution, then X = 1 − Y {\displaystyle 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:[1]

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 ),} {\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)} {\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 ).} {\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).} {\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[2] and Chattamvelli.[1]

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 )}}.} {\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} {\displaystyle B} is the beta function, α {\displaystyle \alpha } {\displaystyle \alpha } and β {\displaystyle \beta } {\displaystyle \beta } are the shape parameters, and λ {\displaystyle \lambda } {\displaystyle \lambda } is the noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.[1]

Transformations

If X ∼ Beta ( α , β , λ ) {\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)} {\displaystyle X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)}, then β X α ( 1 − X ) {\displaystyle {\frac {\beta X}{\alpha (1-X)}}} {\displaystyle {\frac {\beta X}{\alpha (1-X)}}} follows a noncentral F-distribution with 2 α , 2 β {\displaystyle 2\alpha ,2\beta } {\displaystyle 2\alpha ,2\beta } degrees of freedom, and non-centrality parameter λ {\displaystyle \lambda } {\displaystyle \lambda }.

If X {\displaystyle X} {\displaystyle X} follows a noncentral F-distribution F μ 1 , μ 2 ( λ ) {\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)} {\displaystyle F_{\mu _{1},\mu _{2}}\left(\lambda \right)} with μ 1 {\displaystyle \mu _{1}} {\displaystyle \mu _{1}} numerator degrees of freedom and μ 2 {\displaystyle \mu _{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}}}} {\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)} {\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} {\displaystyle \lambda =0}, the noncentral beta distribution is equivalent to the (central) beta distribution.

References

Citations

  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.

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