g01fc returns the deviate associated with the given lower tail probability of the χ2-distribution with real degrees of freedom.

Syntax

C#
public static double g01fc(
	double p,
	double df,
	out int ifail
)
Visual Basic
Public Shared Function g01fc ( _
	p As Double, _
	df As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double g01fc(
	double p, 
	double df, 
	[OutAttribute] int% ifail
)
F#
static member g01fc : 
        p : float * 
        df : float * 
        ifail : int byref -> float 

Parameters

p
Type: System..::..Double
On entry: p, the lower tail probability from the required χ2-distribution.
Constraint: 0.0p<1.0.
df
Type: System..::..Double
On entry: ν, the degrees of freedom of the χ2-distribution.
Constraint: df>0.0.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

g01fc returns the deviate associated with the given lower tail probability of the χ2-distribution with real degrees of freedom.

Description

The deviate, xp, associated with the lower tail probability p of the χ2-distribution with ν degrees of freedom is defined as the solution to
PXxp:ν=p=12ν/2Γν/20xpe-X/2Xv/2-1dX,  0xp<;ν>0.
The required xp is found by using the relationship between a χ2-distribution and a gamma distribution, i.e., a χ2-distribution with ν degrees of freedom is equal to a gamma distribution with scale parameter 2 and shape parameter ν/2.
For very large values of ν, greater than 105, Wilson and Hilferty's normal approximation to the χ2 is used; see Kendall and Stuart (1969).

References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2 distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

Error Indicators and Warnings

Note: g01fc may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
If ifail=123 or 5 on exit, then g01fc returns 0.0.
ifail=1
On entry,p<0.0,
orp1.0.
ifail=2
On entry,df0.0.
ifail=3
p is too close to 0 or 1 for the result to be calculated.
ifail=4
The solution has failed to converge. The result should be a reasonable approximation.
ifail=5
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
ifail=-9000
An error occured, see message report.

Accuracy

The results should be accurate to five significant digits for most parameter values. Some accuracy is lost for p close to 0.0.

Parallelism and Performance

None.

Further Comments

For higher accuracy the relationship described in [Description] may be used and a direct call to g01ff made.

Example

This example reads lower tail probabilities for several χ2-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

Example program (C#): g01fce.cs

Example program data: g01fce.d

Example program results: g01fce.r

See Also