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InvCdfNoncentralF Method
Evaluates the inverse of the noncentral F cumulative distribution function (CDF).

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0
Syntax
Copy
public static double NoncentralF(
	double p,
	double dfn,
	double dfd,
	double lambda
)

Parameters

p
Type: SystemDouble
A double scalar value representing the probability for which the inverse of the noncentral F cumulative distribution function is to be evaluated. p must be non-negative and less than one.
dfn
Type: SystemDouble
A double scalar value representing the number of numerator degrees of freedom. dfn must be positive.
dfd
Type: SystemDouble
A double scalar value representing the number of denominator degrees of freedom. dfd 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 inverse of the noncentral F distribution function evaluated at p. The probability that a noncentral F random variable takes a value less than or equal to InvCdf.NoncentralF(p, dfn, dfd, lambda) is p.
Remarks
If X is a noncentral chi-square random variable with noncentrality parameter \lambda and \nu_1 degrees of freedom, and Y is a chi-square random variable with \nu_2 degrees of freedom which is statistically independent of X, then
F\;\;=\;\;(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)\;\;=\;\; \int_0^f{PDF(x,\nu_1,\nu_2,\lambda)dx}
where the probability density function PDF(x,\nu_1,\nu_2, \lambda) is given by:
PDF(x,\nu_1,\nu_2,\lambda)\;\;=\;\;\Psi\; \sum_{k=0}^\infty{\Phi_k}
\Psi\;\;=\;\;\frac{e^{-\lambda/2}(\nu_1 x)^{\nu_1/2}(\nu_2)^{\nu_2/2}}{x\;(\nu_1 x\;+\;\nu_2)^{(\nu_1+ \nu_2)/2}\;\Gamma(\nu_2/2)}
\Phi_k\;\;=\;\;\frac{R^k\;\Gamma(\frac{ \nu_1+\nu_2}{2}\;+\;k)}{k!\;\Gamma(\frac{\nu_1}{2}\;+\;k)}
R\;\;=\;\;\frac{\lambda\nu_1 x}{2(\nu_1 x \;+\;\nu_2)}
where \Gamma(\cdot) is the Gamma function, \nu_1 = dfn, \nu_2 = dfd, \lambda = lambda, and p\;\;=\;\;CDF(f,\nu_1,\nu_2,\lambda) is the probability that F\le f.

Method InvCdf.NoncentralF evaluates

f\;\;=\;\;CDF^{-1}(p,\nu_1,\nu_2,\lambda)
Method InvCdf.NoncentralF uses bisection and modified regula falsi search algorithms to invert the distribution function CDF(f,\nu_1,\nu_2,\lambda), which is evaluated using method Cdf.NoncentralF. For sufficiently small p, an accurate approximation of CDF^{-1}(p,\nu_1, \nu_2,\lambda) can be used which requires no such inverse search algorithms.
See Also