NAG CL Interface
g01fcc (inv_​cdf_​chisq)

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1 Purpose

g01fcc returns the deviate associated with the given lower tail probability of the χ2-distribution with real degrees of freedom.

2 Specification

#include <nag.h>
double  g01fcc (double p, double df, NagError *fail)
The function may be called by the names: g01fcc, nag_stat_inv_cdf_chisq or nag_deviates_chi_sq.

3 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
P(Xxp:ν)=p=12ν/2Γ(ν/2) 0xpe-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).

4 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

5 Arguments

1: p double Input
On entry: p, the lower tail probability from the required χ2-distribution.
Constraint: 0.0p<1.0.
2: df double Input
On entry: ν, the degrees of freedom of the χ2-distribution.
Constraint: df>0.0.
3: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If fail.code= NE_GAM_NOT_CONV, NE_PROBAB_CLOSE_TO_TAIL, NE_REAL_ARG_GE, NE_REAL_ARG_LE or NE_REAL_ARG_LT on exit, then g01fcc returns 0.0.
On any of the error conditions listed below except fail.code= NE_ALG_NOT_CONV g01fcc returns 0.0.
The algorithm has failed to converge in value iterations. The result should be a reasonable approximation.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
The probability is too close to 0.0 or 1.0.
On entry, p=value.
Constraint: p<1.0.
On entry, df=value.
Constraint: df>0.0.
On entry, p=value.
Constraint: p0.0.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01fcc is not threaded in any implementation.

9 Further Comments

For higher accuracy the relationship described in Section 3 may be used and a direct call to g01ffc made.

10 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.

10.1 Program Text

Program Text (g01fcce.c)

10.2 Program Data

Program Data (g01fcce.d)

10.3 Program Results

Program Results (g01fcce.r)