Noncentral beta distribution
| Notation | Beta(α, β, λ) | ||
|---|---|---|---|
| Parameters |
α > 0 shape (real) β > 0 shape (real) λ ≥ 0 noncentrality (real) | ||
| Support | [math]\displaystyle{ x \in [0; 1]\! }[/math] | ||
| (type I) [math]\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)} }[/math] | |||
| CDF | (type I) [math]\displaystyle{ \sum_{j = 0}^{\infty} e^{-\lambda/2} \frac{\left(\frac{\lambda}{2}\right)^j}{j!} I_x \left(\alpha + j,\beta\right) }[/math] | ||
| Mean | (type I) [math]\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)} {}_2F_2\left(\alpha+\beta,\alpha+1;\alpha,\alpha+\beta+1;\frac{\lambda}{2}\right) }[/math] (see Confluent hypergeometric function) | ||
| Variance | (type I) [math]\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)} {}_2F_2\left(\alpha+\beta,\alpha+2;\alpha,\alpha+\beta+2;\frac{\lambda}{2}\right) - \mu^2 }[/math] where [math]\displaystyle{ \mu }[/math] 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
- [math]\displaystyle{ X = \frac{\chi^2_m(\lambda)}{\chi^2_m(\lambda) + \chi^2_n}, }[/math]
where [math]\displaystyle{ \chi^2_m(\lambda) }[/math] is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter [math]\displaystyle{ \lambda }[/math], and [math]\displaystyle{ \chi^2_n }[/math] is a central chi-squared random variable with degrees of freedom n, independent of [math]\displaystyle{ \chi^2_m(\lambda) }[/math].[1] In this case, [math]\displaystyle{ X \sim \mbox{Beta}\left(\frac{m}{2},\frac{n}{2},\lambda\right) }[/math]
A Type II noncentral beta distribution is the distribution of the ratio
- [math]\displaystyle{ Y = \frac{\chi^2_n}{\chi^2_n + \chi^2_m(\lambda)}, }[/math]
where the noncentral chi-squared variable is in the denominator only.[1] If [math]\displaystyle{ Y }[/math] follows the type II distribution, then [math]\displaystyle{ X = 1 - Y }[/math] 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]
- [math]\displaystyle{ F(x) = \sum_{j=0}^\infty P(j) I_x(\alpha+j,\beta), }[/math]
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 [math]\displaystyle{ I_x(a,b) }[/math] is the incomplete beta function. That is,
- [math]\displaystyle{ F(x) = \sum_{j=0}^\infty \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}I_x(\alpha+j,\beta). }[/math]
The Type II cumulative distribution function in mixture form is
- [math]\displaystyle{ F(x) = \sum_{j=0}^\infty P(j) I_x(\alpha,\beta+j). }[/math]
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:
- [math]\displaystyle{ f(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}\frac{x^{\alpha+j-1}(1-x)^{\beta-1}}{B(\alpha+j,\beta)}. }[/math]
where [math]\displaystyle{ B }[/math] is the beta function, [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are the shape parameters, and [math]\displaystyle{ \lambda }[/math] 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 [math]\displaystyle{ X\sim\mbox{Beta}\left(\alpha,\beta,\lambda\right) }[/math], then [math]\displaystyle{ \frac{\beta X}{\alpha (1-X)} }[/math] follows a noncentral F-distribution with [math]\displaystyle{ 2\alpha, 2\beta }[/math] degrees of freedom, and non-centrality parameter [math]\displaystyle{ \lambda }[/math].
If [math]\displaystyle{ X }[/math] follows a noncentral F-distribution [math]\displaystyle{ F_{\mu_{1}, \mu_{2}}\left( \lambda \right) }[/math] with [math]\displaystyle{ \mu_{1} }[/math] numerator degrees of freedom and [math]\displaystyle{ \mu_{2} }[/math] denominator degrees of freedom, then
- [math]\displaystyle{ Z = \cfrac{\cfrac{\mu_{2}}{\mu_{1}}}{\cfrac{\mu_{2}}{\mu_{1}} + X^{-1} } }[/math]
follows a noncentral Beta distribution:
- [math]\displaystyle{ Z \sim \mbox{Beta}\left(\frac{1}{2}\mu_{1},\frac{1}{2}\mu_{2},\lambda\right) }[/math].
This is derived from making a straightforward transformation.
Special cases
When [math]\displaystyle{ \lambda = 0 }[/math], the noncentral beta distribution is equivalent to the (central) beta distribution.
References
Citations
- ↑ 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. doi:10.1080/00031305.1995.10476151.
- ↑ 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.
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."
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