< 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 . In general, this distribution is unimodal.
The noncentral beta distribution (Type I) is defined as the following ratio :
X = χm2(λ) / (χm2(λ) + χn2),
- 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
- 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 .
One way to evaluate this is with a FORTRAN77 library . Another way to calculate the CDF of the noncentral beta distribution is based on a sharp error bound ; 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  is language independent. However, caution should be used when selecting an algorithm as many return completely incorrect results .
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  and to estimate coil sensitivity profiles in magnetic resonance imaging reconstruction .
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 ASA226 CDF of the noncentral Beta Distribution. Retrieved April 6, 2023 from: https://people.math.sc.edu/Burkardt/f77_src/asa226/asa226.html
 Chattamvelli, R. A Note on the Noncentral Beta Distribution Function. Retrieved April 6, 2023 from: https://www.jstor.org/stable/2684647
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