Cdf.NoncentralF Method

CdfNoncentralF Method

Evaluates the noncentral F cumulative distribution function (CDF).

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

Syntax

public static double NoncentralF(
	double f,
	double df1,
	double df2,
	double lambda
)

Public Shared Function NoncentralF ( 
	f As Double,
	df1 As Double,
	df2 As Double,
	lambda As Double
) As Double

public:
static double NoncentralF(
	double f, 
	double df1, 
	double df2, 
	double lambda
)

static member NoncentralF : 
        f : float * 
        df1 : float * 
        df2 : float * 
        lambda : float -> float

Parameters

f: Type: SystemDouble
A double value representing the argument at which the function is to be evaluated. f must be nonnegative.
df1: Type: SystemDouble
A double value representing the number of numerator degrees of freedom. df1 must be positive.
df2: Type: SystemDouble
A double value representing the number of denominator degrees of freedom. df2 must be positive.
lambda: Type: SystemDouble
A double value representing the noncentrality parameter. lambda must be nonnegative.

Return Value

Type: Double
A double scalar value representing the probability that a noncentral F random variable takes a value less than or equal to f.

Remarks

The noncentral F distribution is a generalization of the F distribution. If is a noncentral chi-square random variable with noncentrality parameter $\lambda$ and $\nu_1$ degrees of freedom, and is a chi-square random variable with $\nu_2$ degrees of freedom which is statistically independent of , then

$F \;\; = \;\; \frac{ (X/\nu_1)}{(Y/\nu_2)}$

is a noncentral F-distributed random variable whose CDF is given by:

$CDF(f, \nu_1, \nu_2, \lambda) \;\; = \;\; \sum_{j = 0}^\infty {c_j}$

where:

$c_j \;\; = \;\; \omega_j \; I_x (\frac{\nu_1}{2} + j, \; \frac{\nu_1}{2})$

$\omega_j \;\; = \;\; e^{-\lambda / 2} \; \frac{(\lambda / 2)^{j}}{j!} \;\; = \;\; \frac{\lambda}{2j} \; \omega_{j-1}$

$I_x (a, b) \;\; = \;\; \frac{B_x (a, b)}{B (a, b)}$

$B_x (a, b) \;\; = \;\; \int_{0}^{x} t^{a-1} (1-t)^{b-1} dt \;\; = \;\; x^{a} \sum_{j = 0}^\infty {\frac{\Gamma(j+1-b)} {(a+j) \; \Gamma(1-b) \; j!}) \; x^{j}}$

$B (a, b) \;\; = \;\; B_1 (a, b) \;\; = \;\; \frac{\Gamma(a) \; \Gamma(b)} {\Gamma(a + b)}$

$I_x (a+1, b) \;\; = \;\; I_x (a, b) \; - \; T_x (a, b)$

$T_x (a, b) \;\; = \;\; \frac{\Gamma(a + b)}{\Gamma(a+1) \; \Gamma(b)} x^{a} (1-x)^{b} \;\; = \;\; T_x (a-1, b) \; \frac{a-1+b}{a} \; x$

$x \;\; = \;\; \frac{\nu_1 f}{\nu_2 \; + \; \nu_1 f}$

and $\Gamma (\cdot)$ is the gamma function, $\nu_1$ = df1, $\nu_2$ = df2, $\lambda$ = lambda, and f = f.

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

Reference

Cdf Class

Imsl.Stat Namespace