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):
tail(j)='L'
The lower tail probability, i.e., pi = P( Xi xpi :νi) .
tail(j)='U'
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.
Constraints:
  • 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
ivalid(i)=0
No error.
ivalid(i)=1
On entry, invalid value supplied in tail when calculating xpi.
ivalid(i)=2
On entry, invalid value for pi.
ivalid(i)=3
On entry, νi0.0.
ivalid(i)=4
pi is too close to 0.0 or 1.0 for the result to be calculated.
ivalid(i)=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)