Pdf.NoncentralBeta Method

PdfNoncentralBeta Method

Evaluates the noncentral beta probability density function.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

public static double NoncentralBeta(
	double x,
	double shape1,
	double shape2,
	double lambda
)

Public Shared Function NoncentralBeta ( 
	x As Double,
	shape1 As Double,
	shape2 As Double,
	lambda As Double
) As Double

public:
static double NoncentralBeta(
	double x, 
	double shape1, 
	double shape2, 
	double lambda
)

static member NoncentralBeta : 
        x : float * 
        shape1 : float * 
        shape2 : float * 
        lambda : float -> float

Parameters

x: Type: SystemDouble
A double scalar value representing the argument at which the function is to be evaluated. x must be nonnegative and less than or equal to 1.
shape1: Type: SystemDouble
A double scalar value representing the first shape parameter. shape1 must be positive.
shape2: Type: SystemDouble
A double scalar value representing the second shape parameter. shape2 must be positive.
lambda: Type: SystemDouble
A double scalar value representing the noncentrality parameter. lambda must nonnegative.

Return Value

Type: Double
A double scalar value representing the probability density associated with a noncentral beta random variable with value x.

Remarks

The noncentral beta distribution is a generalization of the beta distribution. If is a noncentral chi-square random variable with noncentrality parameter $\lambda$ and $2 \alpha_1$ degrees of freedom, and is a chi-square random variable with $2 \alpha_2$ degrees of freedom which is statistically independent of , 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 PDF for noncentral beta variable X can thus be simply defined in terms of the noncentral F PDF:

$PDF_{nc\beta}(x, \; \alpha_1, \; \alpha_2, \; \lambda) \;\; = \;\; PDF_{ncF}(f, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda) \; \frac{df}{dx}$

where $PDF_{nc\beta}(x, \; \alpha_1, \; \alpha_2, \; \lambda)$ is the noncentral beta PDF with = x, $\alpha_1$ = shape1, $\alpha_2$ = shape2, and noncentrality parameter $\lambda$ = lambda; $PDF_{ncF}(f, \; 2 \alpha_1, \; 2 \alpha_2, \; \lambda)$ is the noncentral F PDF with argument f, numerator and denominator degrees of freedom $2 \alpha_1$ and $2 \alpha_2$ respectively, noncentrality parameter $\lambda$ ,

$f \;\; = \;\; \frac{\alpha_2 x}{\alpha_1 (1 \; - \; x)},$

$x \;\; = \;\; \frac{\alpha_1 f}{\alpha_1 f \; + \; \alpha_2},$

and

$\frac{df}{dx} \;\; = \;\; \frac{(\alpha_1 f \; + \; \alpha_2)^2}{\alpha_1 \alpha_2} \;\; = \;\; \frac{\alpha_2}{\alpha_1 (1 \; - \; x)^2}.$

(See documentation for class Pdf method NoncentralF for a discussion of how the noncentral F PDF is defined and calculated.)

With a noncentrality parameter of zero, the noncentral beta distribution is the same as the beta distribution.

Reference

Pdf Class

Imsl.Stat Namespace