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\in [0;1]\!
pdf (type I) \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) \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^{{-{\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^{{-{\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 \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 generalization of the (central) beta distribution.

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

X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},

where \chi _{m}^{2}(\lambda ) is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter \lambda , and \chi _{n}^{2} is a central chi-squared random variable with degrees of freedom n, independent of \chi _{m}^{2}(\lambda ).[1] 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 _{n}^{2}}{\chi _{n}^{2}+\chi _{m}^{2}(\lambda )}},

where the noncentral chi-squared variable is in the denominator only.[1] 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:[1]

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)^{j}e^{{-\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[2] and Chattamvelli.[1]

Probability density function

The (Type I) probability density function for the noncentral beta distribution is:

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 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.[1]

Related distributions

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 .

Special cases

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

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician 49 (2): 231–234 url=http://amstat.tandfonline.com/doi/abs/10.1080/00031305.1995.10476151#.UhpzB3_3Oy8. 
  2. ↑ Posten, H.O. (1993). "An Effective Algorithm for the Noncentral Beta Distribution Function". The American Statistician 47 (2): 129–131. JSTOR 2685195. 
  • 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: 648–653. 
  • Seber, G.A.F. (1963). "The non-central chi-squared and beta distributions". Biometrika 50: 542–544. 
  • Christian Walck, "Hand-book on Statistical Distributions for experimentalists."
 
Continuous univariate supported on a bounded interval, e.g. [0,1]
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