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

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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: tail(ltail) Character(1) array Input
On entry: indicates which tail the supplied probabilities represent. For j= ((i-1) mod ltail) +1 , for i=1,2,,max(ltail,lp,ldf):
The lower tail probability, i.e., pi = P( Xi xpi :νi) .
The upper tail probability, i.e., pi = P( Xi xpi :νi) .
Constraint: tail(j)='L' or 'U', for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: p(lp) Real (Kind=nag_wp) array Input
On entry: pi, the probability of the required χ2-distribution as defined by tail with pi=p(j), j=((i-1) mod lp)+1.
  • if tail(k)='L', 0.0p(j)<1.0;
  • otherwise 0.0<p(j)1.0.
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: df(ldf) Real (Kind=nag_wp) array Input
On entry: νi, the degrees of freedom of the χ2-distribution with νi=df(j), j=((i-1) mod ldf)+1.
Constraint: df(j)>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 max(ltail,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 max(ltail,lp,ldf).
On exit: ivalid(i) indicates any errors with the input arguments, with
No error.
On entry, invalid value supplied in tail when calculating xpi.
On entry, invalid value for pi.
On entry, νi0.0.
pi is too close to 0.0 or 1.0 for the result to be calculated.
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:
On entry, at least one value of tail, p or df was invalid, or the solution failed to converge.
Check ivalid for more information.
On entry, array size=value.
Constraint: ltail>0.
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: ldf>0.
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.
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.
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

Background information to multithreading can be found in the Multithreading documentation.
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)