Evaluates the inverse of the noncentral beta cumulative distribution function (CDF).

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
public static double NoncentralBeta(
	double p,
	double shape1,
	double shape2,
	double lambda
)
Visual Basic (Declaration)
Public Shared Function NoncentralBeta ( _
	p As Double, _
	shape1 As Double, _
	shape2 As Double, _
	lambda As Double _
) As Double
Visual C++
public:
static double NoncentralBeta(
	double p, 
	double shape1, 
	double shape2, 
	double lambda
)

Parameters

p
Type: System..::.Double
A double scalar value representing the probability for which the inverse of the noncentral beta cumulative distribution function is to be evaluated. p must be non-negative and less than or equal to one.
shape1
Type: System..::.Double
A double scalar value representing the first shape parameter. shape1 must be positive.
shape2
Type: System..::.Double
A double scalar value representing the second shape parameter. shape2 must be positive.
lambda
Type: System..::.Double
A double scalar value representing the noncentrality parameter. lambda must be nonnegative.

Return Value

A double scalar value representing the inverse of the noncentral beta distribution function evaluated at p. The probability that a noncentral beta random variable takes a value less than or equal to NoncentralBeta is p.

Remarks

If Z is a noncentral chi-square random variable with noncentrality parameter \lambda and 2\alpha_1 degrees of freedom, and Y is a chi-square random variable with 2\alpha_2 degrees of freedom which is statistically independent of Z, then
X\;\;=\;\;\frac{Z}{Z\;+\;Y}\;\;=\;\; \frac{\alpha_1 F}{\alpha_1 F\;+\;\alpha_2}
is a noncentral beta-distributed random variable and
F\;\;=\;\;\frac{\alpha_2 Z}{\alpha_1 Y}\; \;=\;\;\frac{\alpha_2 X}{\alpha_1(1\;-\;X)}
is a noncentral F-distributed random variable. The CDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F CDF:
CDF_{nc\beta}(x,\;\alpha_1,\;\alpha_2,\; \lambda)\;\;=\;\;CDF_{ncF}(f,\;2\alpha_1,\;2\alpha_2,\;\lambda)
where CDF_{nc\beta}(x,\;\alpha_1,\;\alpha_2,\;\lambda) is the noncentral beta CDF with x = x, \alpha_1 = shape1, \alpha_2 = shape2, and noncentrality parameter \lambda = lambda; CDF_{ncF} (f,\;2\alpha_1,\;2\alpha_2,\;\lambda) is the noncentral F CDF with argument f, numerator and denominator degrees of freedom 2\alpha_1 and 2\alpha_2 respectively, and noncentrality parameter \lambda; and:
f\;\;=\;\;\frac{\alpha_2 x}{\alpha_1(1\;- \;x)};\;\;x\;\;=\;\;\frac{\alpha_1 f}{\alpha_1 f\;+\;\alpha_2}
(See documentation for class Cdf method NoncentralF for a discussion of how the noncentral F CDF is defined and calculated.)

Method InvCdf.NoncentralBeta evaluates

x\;\;=\;\;CDF_{nc\beta}^{-1}(p,\; \alpha_1,\;\alpha_2,\;\lambda)
by first evaluating:
f\;\;=\;\;CDF_{ncF}^{-1}(p,\;2\alpha_1,\; 2\alpha_2,\;\lambda)
and then solving for x using x\;\;=\;\;\frac{ \alpha_1 f}{\alpha_1 f\;+\;\alpha_2}. (See documentation for InvCdf.NoncentralF for a discussion of how the inverse noncentral F CDF is calculated.)

See Also