g01fc returns the deviate associated with the given lower tail probability of the

χ^{2}

-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 $χ^{2}$ -distribution.

Constraint: $0.0 \leq p < 1.0$ .

df: Type: System..::..Double
On entry: $ν$ , the degrees of freedom of the $χ^{2}$ -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

χ^{2}

-distribution with real degrees of freedom.

Description

The deviate,

x_{p}

, associated with the lower tail probability

p

of the

χ^{2}

-distribution with

ν

degrees of freedom is defined as the solution to

P (X \leq x_{p} : ν) = p = \frac{1}{2^{ν / 2} Γ (ν / 2)} \int_{0}^{x_{p}} e^{- X / 2} X^{v / 2 - 1} d X, 0 \leq x_{p} < \infty; ν > 0 .

The required

x_{p}

is found by using the relationship between a

χ^{2}

-distribution and a gamma distribution, i.e., a

χ^{2}

-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

10^{5}

, Wilson and Hilferty's normal approximation to the

χ^{2}

is used; see Kendall and Stuart (1969).

References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the

χ^{2}

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:

ifail = 1

2

3

5

on exit, then g01fc returns

0.0

$ifail = 1$

On entry,	$p < 0.0$ ,
or	$p \geq 1.0$ .

$ifail = 2$

On entry,

df \leq 0.0

$ifail = 3$: 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

χ^{2}

-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