# Noncentral beta distribution

< List of probability distributions < Noncentral beta distribution

The Noncentral Beta Distribution, a generalization of the beta distribution, surpasses the limitations of the “usual” (or centralized) beta distribution by modeling data associated with values next to zero and one [1]. In general, this distribution is unimodal.

The noncentral beta distribution (Type I) is defined as the following ratio [2]:

X = χm2(λ) / (χm2(λ) + χn2),

where

• χm2(λ) = a noncentral χ2 (chi-squared) random variable with m degrees of freedom, and
• χn2 = a central χ2 random variable with n degrees of freedom.

The Type II noncentral beta distribution is defined as the ratio

X = χn2 / (χn2 + χm2(λ)).

The two types are related in the following way: If a random variable Y follows a type II distribution, then X = 1 – Y follows a type I distribution.

## Noncentral Beta Distribution Properties

While there have been many different parameterizations of the noncentral beta distribution since its derivation in the 1920s, one version of the probability density function (PDF) is [3]

Where:

• β, γ = positive shape parameters,
• δ = positive noncentrality parameter,
• B = the beta function.

The cumulative distribution function (CDF), which is intractable (very difficult to solve), requires significant digit computation of both the Poisson distribution and the central beta distributions [4].

One way to evaluate this is with a FORTRAN77 library [5]. Another way to calculate the CDF of the noncentral beta distribution is based on a sharp error bound [6]; sharp error bounds guarantee that the actual value of the function being approximated is within a certain distance of the approximation. The sharp error bound for the noncentral beta distribution is based on a theorem called the saddlepoint approximation. Various other means of calculating the CDF have been proposed, most of which have been formulated for various programming languages but Posten’s step-by-step algorithm [7] is language independent. However, caution should be used when selecting an algorithm as many return completely incorrect results [4].

For any random variable X, calculating other statistical measures such as population variance, skewness, and kurtosis can also be quite challenging. These measures are not impossible to calculate, but — like the CDF — they require complex algorithms or specialized tools.

## Applications of the noncentral beta distribution

The noncentral beta distribution may have limitations in terms of interpretability and tractability, but it has some practical applications that make it worth exploring. For example, it has been used to model a single wave propagating amongst noise in a receiver array [8] and to estimate coil sensitivity profiles in magnetic resonance imaging reconstruction [9].

## References

[1] Orsi, C. (2017). New insights into non-central beta distributions. Retrieved November 29, 2021 from: https://arxiv.org/pdf/1706.08557.pdf

[2] Chattamvelli, R. A note on the Noncentral Beta Distribution Function.

[3] Noncentral beta distribution. Retrieved April 6, 2023 from: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Noncentralbeta.pdf

[4] Baharev, A. & Kemeny, S. On the computation of the noncentral F and noncentral beta distribution. Statistics and Computing, 2008, 18 (3), 333-340. Springer.

[5] ASA226 CDF of the noncentral Beta Distribution. Retrieved April 6, 2023 from: https://people.math.sc.edu/Burkardt/f77_src/asa226/asa226.html

[6] Chattamvelli, R. A Note on the Noncentral Beta Distribution Function. Retrieved April 6, 2023 from: https://www.jstor.org/stable/2684647

[7] Kimball, C.V., Scheibner, D.J.: Error bars for sonic slowness measurements. Geophysics, 63, 345–353, (1998)

[8] Stamm, A., Singh, J., Afacan, O., Warfield, S. K.: Analytic quantification of biasand variance of coil sensitivity profile estimators for improved image reconstructionin MRI. Medical Image Computing and Computer-Assisted Intervention MICCAI 2015, 684–691 (2015)

[9] Posten (1993). An Effective Algorithm for the Noncentral Beta Distribution Function. The American Statistician Vol. 47, No. 2 (May, 1993), pp. 129-131 (3 pages) Published By: Taylor & Francis, Ltd.

Scroll to Top