NAG FL Interface
g01tdf (inv_​cdf_​f_​vector)

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1 Purpose

g01tdf returns a number of deviates associated with given probabilities of the F or variance-ratio distribution with real degrees of freedom.

2 Specification

Fortran Interface
Subroutine g01tdf ( ltail, tail, lp, p, ldf1, df1, ldf2, df2, f, ivalid, ifail)
Integer, Intent (In) :: ltail, lp, ldf1, ldf2
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: ivalid(*)
Real (Kind=nag_wp), Intent (In) :: p(lp), df1(ldf1), df2(ldf2)
Real (Kind=nag_wp), Intent (Out) :: f(*)
Character (1), Intent (In) :: tail(ltail)
C Header Interface
#include <nag.h>
void  g01tdf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *ldf1, const double df1[], const Integer *ldf2, const double df2[], double f[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01tdf or nagf_stat_inv_cdf_f_vector.

3 Description

The deviate, fpi, associated with the lower tail probability, pi, of the F-distribution with degrees of freedom ui and vi is defined as the solution to
P( Fi fpi :ui,vi) = pi = u i 12 ui v i 12 vi Γ ( ui + vi 2 ) Γ ( ui 2 ) Γ ( vi 2 ) 0 fpi Fi 12 (ui-2) (vi+uiFi) -12 (ui+vi) dFi ,  
where ui,vi>0; 0fpi<.
The value of fpi is computed by means of a transformation to a beta distribution, P iβi ( Bi βi :ai,bi) :
P( Fi fpi :ui,vi) = P iβi ( Bi ui fpi ui fpi + vi : ui / 2 , vi / 2 )  
and using a call to g01tef.
For very large values of both ui and vi, greater than 105, a Normal approximation is used. If only one of ui or vi is greater than 105 then a χ2 approximation is used; see Abramowitz and Stegun (1972).
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

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

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,ldf1,ldf2):
The lower tail probability, i.e., pi = P( Fi fpi : ui , vi ) .
The upper tail probability, i.e., pi = P( Fi fpi : ui , vi ) .
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 F-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: ldf1 Integer Input
On entry: the length of the array df1.
Constraint: ldf1>0.
6: df1(ldf1) Real (Kind=nag_wp) array Input
On entry: ui, the degrees of freedom of the numerator variance with ui=df1(j), j=((i-1) mod ldf1)+1.
Constraint: df1(j)>0.0, for j=1,2,,ldf1.
7: ldf2 Integer Input
On entry: the length of the array df2.
Constraint: ldf2>0.
8: df2(ldf2) Real (Kind=nag_wp) array Input
On entry: vi, the degrees of freedom of the denominator variance with vi=df2(j), j=((i-1) mod ldf2)+1.
Constraint: df2(j)>0.0, for j=1,2,,ldf2.
9: f(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array f must be at least max(ltail,lp,ldf1,ldf2).
On exit: fpi, the deviates for the F-distribution.
10: ivalid(*) Integer array Output
Note: the dimension of the array ivalid must be at least max(ltail,lp,ldf1,ldf2).
On exit: ivalid(i) indicates any errors with the input arguments, with
No error.
On entry, invalid value supplied in tail when calculating fpi.
On entry, invalid value for pi.
On entry, ui0.0, or, vi0.0.
The solution has not converged. The result should still be a reasonable approximation to the solution.
The value of pi is too close to 0.0 or 1.0 for the result to be computed. This will only occur when the large sample approximations are used.
11: 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 −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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:
Note: in some cases g01tdf may return useful information.
On entry, at least one value of tail, p, df1, df2 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: ldf1>0.
On entry, array size=value.
Constraint: ldf2>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 result should be accurate to five significant digits.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01tdf is not threaded in any implementation.

9 Further Comments

For higher accuracy g01tef can be used along with the transformations given in Section 3.

10 Example

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

10.1 Program Text

Program Text (g01tdfe.f90)

10.2 Program Data

Program Data (g01tdfe.d)

10.3 Program Results

Program Results (g01tdfe.r)