Supriya Ghosh (Editor)

Noncentral beta distribution

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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]\!}

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)}}}

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)}

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)} (see Confluent hypergeometric function)

In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.

Contents

The noncentral beta distribution (Type I) is the distribution of the ratio

X = χ m 2 ( λ ) χ m 2 ( λ ) + χ n 2 ,

where χ m 2 ( λ ) is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter λ , and χ n 2 is a central chi-squared random variable with degrees of freedom n, independent of χ m 2 ( λ ) . In this case, X ∼ Beta ( m 2 , n 2 , λ )

A Type II noncentral beta distribution is the distribution of the ratio

Y = χ n 2 χ n 2 + χ m 2 ( λ ) ,

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 ) = ∑ j = 0 ∞ P ( j ) I x ( α + j , β ) ,

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 ) = ∑ j = 0 ∞ 1 j ! ( λ 2 ) j e − λ / 2 I x ( α + j , β ) .

The Type II cumulative distribution function in mixture form is

F ( x ) = ∑ j = 0 ∞ P ( j ) I x ( α , β + 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 ) = ∑ j = 0 ∞ 1 j ! ( λ 2 ) j e − λ / 2 x α + j − 1 ( 1 − x ) β − 1 B ( α + j , β ) .

where B is the beta function, α and β are the shape parameters, and λ 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 ∼ Beta ( α , β , λ ) , then β X α ( 1 − X ) follows a noncentral F-distribution with 2 α , 2 β degrees of freedom, and non-centrality parameter λ .

If X follows a noncentral F-distribution F μ 1 , μ 2 ( λ ) with μ 1 numerator degrees of freedom and μ 2 denominator degrees of freedom, then Z = μ 2 μ 1 μ 2 μ 1 + X − 1 follows a noncentral Beta distribution so Z ∼ Beta ( 1 2 μ 1 , 1 2 μ 2 , λ ) . This is derived from making a straightforward transformation.

Special cases

When λ = 0 , the noncentral beta distribution is equivalent to the (central) beta distribution.

References

Noncentral beta distribution Wikipedia
(Text) CC BY-SA


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