NAG CL Interface
g01tdc (inv_​cdf_​f_​vector)

1 Purpose

g01tdc returns a number of deviates associated with given probabilities of the F or variance-ratio distribution with real degrees of freedom.

2 Specification

#include <nag.h>
void  g01tdc (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer ldf1, const double df1[], Integer ldf2, const double df2[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: g01tdc, nag_stat_inv_cdf_f_vector or nag_deviates_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 + ui Fi -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 g01tec.
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 function 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] const Nag_TailProbability Input
On entry: indicates which tail the supplied probabilities represent. For j= i-1 mod ltail , for i=1,2,,maxltail,lp,ldf1,ldf2:
tail[j]=Nag_LowerTail
The lower tail probability, i.e., pi = P Fi fpi : ui , vi .
tail[j]=Nag_UpperTail
The upper tail probability, i.e., pi = P Fi fpi : ui , vi .
Constraint: tail[j-1]=Nag_LowerTail or Nag_UpperTail, for j=1,2,,ltail.
3: lp Integer Input
On entry: the length of the array p.
Constraint: lp>0.
4: p[lp] const double Input
On entry: pi, the probability of the required F-distribution as defined by tail with pi=p[j], j=i-1 mod lp.
Constraints:
  • if tail[k]=Nag_LowerTail, 0.0p[j]<1.0;
  • otherwise 0.0<p[j]1.0.
Where k=i-1 mod ltail and j=i-1 mod lp.
5: ldf1 Integer Input
On entry: the length of the array df1.
Constraint: ldf1>0.
6: df1[ldf1] const double Input
On entry: ui, the degrees of freedom of the numerator variance with ui=df1[j], j=i-1 mod ldf1.
Constraint: df1[j-1]>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] const double Input
On entry: vi, the degrees of freedom of the denominator variance with vi=df2[j], j=i-1 mod ldf2.
Constraint: df2[j-1]>0.0, for j=1,2,,ldf2.
9: f[dim] double Output
Note: the dimension, dim, of the array f must be at least maxltail,lp,ldf1,ldf2.
On exit: fpi, the deviates for the F-distribution.
10: ivalid[dim] Integer Output
Note: the dimension, dim, of the array ivalid must be at least maxltail,lp,ldf1,ldf2.
On exit: ivalid[i-1] indicates any errors with the input arguments, with
ivalid[i-1]=0
No error.
ivalid[i-1]=1
On entry, invalid value supplied in tail when calculating fpi.
ivalid[i-1]=2
On entry, invalid value for pi.
ivalid[i-1]=3
On entry, ui0.0, or, vi0.0.
ivalid[i-1]=4
The solution has not converged. The result should still be a reasonable approximation to the solution.
ivalid[i-1]=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: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, array size=value.
Constraint: ldf1>0.
On entry, array size=value.
Constraint: ldf2>0.
On entry, array size=value.
Constraint: lp>0.
On entry, array size=value.
Constraint: ltail>0.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of tail, p, df1, df2 was invalid, or the solution failed to converge.
Check ivalid for more information.

7 Accuracy

The result should be accurate to five significant digits.

8 Parallelism and Performance

g01tdc is not threaded in any implementation.

9 Further Comments

For higher accuracy g01tec 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 (g01tdce.c)

10.2 Program Data

Program Data (g01tdce.d)

10.3 Program Results

Program Results (g01tdce.r)