Non-central Beta Distribution

The non-central Beta distribution is a generalization of the Beta distribution. It is characterized by two shape parameters, α and β, and a non-centrality parameter. It is also known as the non-central Beta prime distribution.

Definition

The non-central Beta distribution has shape parameters α, β > 0 and non-centrality parameter λ ≥ 0. Its probability density function (PDF) is:

f (x; α, β, λ) = e^{- λ / 2} \sum_{j = 0}^{\infty} \frac{(λ / 2)^{j}}{j!} \frac{x^{α + j - 1} (1 - x)^{β - 1}}{B (α + j, β)}

The cumulative distribution function (CDF) can be expressed as an infinite series:

F (x; α, β, λ) = e^{- λ / 2} \sum_{j = 0}^{\infty} \frac{(λ / 2)^{j}}{j!} I_{x} (α + j, β)

where $I_{x} (a, b)$ is the regularized incomplete beta function and $x \in [0, 1]$ .

Applications

The non-central Beta distribution finds extensive applications across multiple scientific and engineering disciplines.

In Bayesian statistics, the distribution serves as a posterior distribution for probability parameters when analyzing binomial data with prior information.
Quality control processes utilize this distribution to model the variation in manufacturing tolerances when there is a systematic bias present.
Reliability engineers employ the non-central Beta distribution to analyze system lifetime data where component failures follow non-symmetric patterns.
In clinical trials, researchers use this distribution to model the success rates of treatments when there are underlying systematic effects.

Properties

Statistical Properties

Property	Value
Mean	$\frac{α (α + β + λ)}{α + β}$
Variance	$\frac{α β (α + β + λ)}{(α + β)^{2} (α + β + 1)}$
Mode	No closed form

Notable properties include:

The distribution is supported on the interval [0,1].
Higher moments involve confluent hypergeometric functions.
The distribution is not symmetric unless α = β and λ = 0.

Relationships to Other Distributions

When the non-centrality parameter λ = 0, the distribution reduces to the standard Beta distribution.
The distribution is related to the non-central F-distribution through a transformation.
For large shape parameters (α, β), the distribution approximates a non-central normal distribution.

Several special cases exist for specific parameter values:

When α = β = 1 and λ = 0, the distribution is equivalent to the uniform distribution.
When (α = 1, β = 2) or (α = 2, β = 1) and λ = 0, the distribution reduces to a triangular distribution.

The NonCentralBetaDistribution Class

The non-central Beta distribution is implemented by the NonCentralBetaDistribution class. It has one constructor that takes three arguments: the two shape parameters, α and β, followed by the non-centrality parameter. The following constructs a non-central Beta distribution with α = 1.5, β = 0.8, and non-centrality parameter 2:

var ncBeta1 = new NonCentralBetaDistribution(1.5, 0.8, 2.0);

Visual Basic

Dim ncBeta1 = New NonCentralBetaDistribution(1.5, 0.8, 2.0)

Visual Basic

No code example is currently available or this language may not be supported.

let ncBeta1 = NonCentralBetaDistribution(1.5, 0.8, 2.0)

The NonCentralBetaDistribution class has three specific properties that correspond to the parameters of the distribution. The Alpha and Beta properties return the shape parameters, α and β. The NonCentralityParameter property returns the non-centrality parameter.

References

Wikipedia: Noncentral Beta Distribution
MathWorld: Noncentral Beta Distribution