Noncentral Beta Distribution

Probability Distributions > Noncentral beta distribution

The noncentral beta distribution is a generalization of the Beta Distribution. The Beta distribution, although useful in many ways, has a significant drawback: it cannot properly model portions of data that have values next to zero and one [1]. The noncentral beta distribution overcomes this drawback. Some practical applications of the distribution include:

• Modeling single wave propagating across a receiver array with added Gaussian noise [2].
• Estimating coil sensitivity profiles in magnetic resonance image reconstruction [3].

Noncentral beta distribution properties

The noncentral beta distribution (Type I) is defined as the following ratio [4]:

X = χ_m²(λ) / (χ_m²(λ) + χ_n²),

where

• χ_m²(λ) = a noncentral χ² (chi-squared) random variable with m degrees of freedom, and
• χ_n² = a central χ² random variable with n degrees of freedom.

The Type II noncentral beta distribution is defined as the ratio

X = χ_n² / (χ_n² + 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. A noncentral beta distribution has a probability density function (PDF) of
Where:

• β and γ are positive shape parameters,
• δ is a positive noncentrality parameter.

In general, the PDF is unimodal with a single peak. Note that there have been many different parametrizations of the noncentral beta distribution since its derivation in the 1920s.

The noncentral beta distribution has some strong limitations in terms of interpretability and tractability. For any random variable X with a noncentral beta distribution, the following are mathematically intractable [4]:

• Cumulative distribution function,
• Survivor function,
• Hazard and cumulative hazard function,
• Inverse distribution,
• Moment generating function,
• Characteristic functions,
• Population mean,
• Population variance,
• Skewness,
• Kurtosis.

Note that intractable means difficult or hard, not “impossible.” There are ways to calculate them, although they are complicated. For example, the CDF can be calculated with various algorithms, the simplest of which is based on a sharp error bound [5]. 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 [6] is language independent. However, caution should be used when selecting an algorithm as many return completely incorrect results [4]. A FORTRAN77 library is also available to evaluate the CDF.  

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 [7] and to estimate coil sensitivity profiles in magnetic resonance imaging reconstruction [8].

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

[3] 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)

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

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

[7] 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)

[8] 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.