Noncentral beta distribution

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) $e^{-\frac{\lambda}{2}}\frac{\Gamma\left(\alpha + 1\right)}{\Gamma\left(\alpha\right)} \frac{\Gamma\left(\alpha+\beta\right)}{\Gamma\left(\alpha + \beta + 1\right)} {}_2F_2\left(\alpha+\beta,\alpha+1;\alpha,\alpha+\beta+1;\frac{\lambda}{2}\right)$ (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 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 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 $\chi^2_m(\lambda)$ .^[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^2_n}{\chi^2_n + \chi^2_m(\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)^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^[2] and Chattamvelli.^[1]

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.^[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 $\lambda$ .

If $X$ follows a noncentral F-distribution $F_{\mu_{1}, \mu_{2}}\left( \lambda \right)$ with $\mu _{{1}}$ numerator degrees of freedom and $\mu _{{2}}$ denominator degrees of freedom, then $Z = \cfrac{\cfrac{\mu_{2}}{\mu_{1}}}{\cfrac{\mu_{2}}{\mu_{1}} + X^{-1} }$ follows a noncentral Beta distribution so $Z \sim \mbox{Beta}\left(\frac{1}{2}\mu_{1},\frac{1}{2}\mu_{2},\lambda\right)$ . This is derived from making a straightforward transformation.

Special cases

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

References

1 2 3 4 5 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. 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. doi:10.1214/aoms/1177728424.
Seber, G.A.F. (1963). "The non-central chi-squared and beta distributions". Biometrika. 50: 542–544. doi:10.1093/biomet/50.3-4.542.
Christian Walck, "Hand-book on Statistical Distributions for experimentalists."

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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