Evaluates the inverse of the chi-squared cumulative probability distribution function.

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

Syntax

C#
public static double Chi(
	double p,
	double df
)
Visual Basic (Declaration)
Public Shared Function Chi ( _
	p As Double, _
	df As Double _
) As Double
Visual C++
public:
static double Chi(
	double p, 
	double df
)

Parameters

p
Type: System..::.Double
A double scalar value representing the probability for which the inverse chi-squared function is to be evaluated.
df
Type: System..::.Double
A double scalar value representing the number of degrees of freedom. This must be at least 0.5.

Return Value

A double scalar value. The probability that a chi-squared random variable takes a value less than or equal to this returned value is p.

Remarks

Method InvCdf.Chi evaluates the inverse distribution function of a chi-squared random variable with df degrees of freedom, that is, with P = p and v = df, it determines x (equal to InvCdf.Chi(p, df)), such that
P=\frac{1}{{2^{\nu/2}\Gamma\left({\nu/2} \right)}}\int_0^x{e^{-t/2}t^{\nu/2-1}}dt
where \Gamma(\cdot) is the Gamma function. The probability that the random variable takes a value less than or equal to x is P.

For v\lt 40, InvCdf.Chi uses bisection, if v\ge 2 or P\gt 0.98, or regula falsi to find the point at which the chi-squared distribution function is equal to P. The distribution function is evaluated using Cdf.Chi.

For 40\le v\lt 100, a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.18) to the normal distribution is used, and InvCdf.Normal is used to evaluate the inverse of the normal distribution function. For v\ge 100, the ordinary Wilson-Hilferty approximation (Abramowitz and Stegun 1964, equation 26.4.17) is used.

See Also