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Noncentral beta distribution

Probability distribution

Probability distribution

name =Noncentral Beta| type = density| 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)} {}_2F_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)} {}_2F_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 noncentral generalization of the (central) beta distribution.

The noncentral beta distribution (Type I) is the distribution of the ratio : X = \frac{\chi^2_m(\lambda)}{\chi^2_m(\lambda) + \chi^2_n},

where \chi^2_m(\lambda) is a [[Noncentral chi-squared distribution| noncentral chi-squared]] random variable with degrees of freedom m and noncentrality parameter \lambda, and \chi^2_n is a central chi-squared random variable with degrees of freedom n, independent of \chi^2_m(\lambda).{{cite journal 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^2_n}{\chi^2_n + \chi^2_m(\lambda)}, 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) = \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)^je^{-\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 and Chattamvelli.

Probability density function

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

: 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)}.

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.

References

Citations

Sources

  • M. Abramowitz and I. Stegun, editors (1965) "Handbook of Mathematical Functions", Dover: New York, NY.
  • Christian Walck, "Hand-book on Statistical Distributions for experimentalists."

References

  1. Posten, H.O.. (1993). "An Effective Algorithm for the Noncentral Beta Distribution Function". The American Statistician.
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