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):
tail(j)='L'
The lower tail probability, i.e., pi = P( Fi fpi : ui , vi ) .
tail(j)='U'
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.
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: 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
ivalid(i)=0
No error.
ivalid(i)=1
On entry, invalid value supplied in tail when calculating fpi.
ivalid(i)=2
On entry, invalid value for pi.
ivalid(i)=3
On entry, ui0.0, or, vi0.0.
ivalid(i)=4
The solution has not converged. The result should still be a reasonable approximation to the solution.
ivalid(i)=5
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.
ifail=1
On entry, at least one value of tail, p, df1, df2 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: ldf1>0.
ifail=5
On entry, array size=value.
Constraint: ldf2>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 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)