NAG FL Interface
g01tcf (inv_​cdf_​chisq_​vector)

1 Purpose

g01tcf returns a number of deviates associated with the given probabilities of the χ2-distribution with real degrees of freedom.

2 Specification

Fortran Interface
Subroutine g01tcf ( ltail, tail, lp, p, ldf, df, x, ivalid, ifail)
Integer, Intent (In) :: ltail, lp, ldf
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(*)
Real (Kind=nag_wp), Intent (In) :: p(lp), df(ldf)
Real (Kind=nag_wp), Intent (Out) :: x(*)
Character (1), Intent (In) :: tail(ltail)
C Header Interface
#include <nag.h>
void  g01tcf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *ldf, const double df[], double x[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01tcf or nagf_stat_inv_cdf_chisq_vector.

3 Description

The deviate, xpi, associated with the lower tail probability pi of the χ2-distribution with νi degrees of freedom is defined as the solution to
P Xi xpi :νi = pi = 1 2 νi/2 Γ νi/2 0 xpi e -Xi/2 Xi vi / 2 - 1 dXi ,   0 xpi < ; ​ νi > 0 .  
The required xpi is found by using the relationship between a χ2-distribution and a gamma distribution, i.e., a χ2-distribution with νi degrees of freedom is equal to a gamma distribution with scale parameter 2 and shape parameter νi/2.
For very large values of νi, greater than 105, Wilson and Hilferty's Normal approximation to the χ2 is used; see Kendall and Stuart (1969).
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

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: ltail Integer Input
On entry: the length of the array tail.
Constraint: ltail>0.
2: tailltail Character(1) array Input
On entry: indicates which tail the supplied probabilities represent. For j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,ldf:
tailj='L'
The lower tail probability, i.e., pi = P Xi xpi :νi .
tailj='U'
The upper tail probability, i.e., pi = P Xi xpi :νi .
Constraint: tailj='L' or 'U', for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: plp Real (Kind=nag_wp) array Input
On entry: pi, the probability of the required χ2-distribution as defined by tail with pi=pj, j=i-1 mod lp+1.
Constraints:
  • if tailk='L', 0.0pj<1.0;
  • otherwise 0.0<pj1.0.
Where k=i-1 mod ltail+1 and j=i-1 mod lp+1.
5: ldf Integer Input
On entry: the length of the array df.
Constraint: ldf>0.
6: dfldf Real (Kind=nag_wp) array Input
On entry: νi, the degrees of freedom of the χ2-distribution with νi=dfj, j=i-1 mod ldf+1.
Constraint: dfj>0.0, for j=1,2,,ldf.
7: x* Real (Kind=nag_wp) array Output
Note: the dimension of the array x must be at least maxltail,lp,ldf.
On exit: xpi, the deviates for the χ2-distribution.
8: ivalid* Integer array Output
Note: the dimension of the array ivalid must be at least maxltail,lp,ldf.
On exit: ivalidi indicates any errors with the input arguments, with
ivalidi=0
No error.
ivalidi=1
On entry, invalid value supplied in tail when calculating xpi.
ivalidi=2
On entry, invalid value for pi.
ivalidi=3
On entry, νi0.0.
ivalidi=4
pi is too close to 0.0 or 1.0 for the result to be calculated.
ivalidi=5
The solution has failed to converge. The result should be a reasonable approximation.
9: ifail Integer Input/Output
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.
ifail=2
On entry, array size=value.
Constraint: ltail>0.
ifail=3
On entry, array size=value.
Constraint: lp>0.
ifail=4
On entry, array size=value.
Constraint: ldf>0.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The results should be accurate to five significant digits for most argument values. Some accuracy is lost for pi close to 0.0 or 1.0.

8 Parallelism and Performance

g01tcf 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 g01tff made.

10 Example

This example reads lower tail probabilities for several χ2-distributions, and calculates and prints the corresponding deviates.

10.1 Program Text

Program Text (g01tcfe.f90)

10.2 Program Data

Program Data (g01tcfe.d)

10.3 Program Results

Program Results (g01tcfe.r)