NAG CL Interface
g01fdc (inv_​cdf_​f)

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1 Purpose

g01fdc returns the deviate associated with the given lower tail probability of the F or variance-ratio distribution with real degrees of freedom.

2 Specification

#include <nag.h>
double  g01fdc (double p, double df1, double df2, NagError *fail)
The function may be called by the names: g01fdc, nag_stat_inv_cdf_f or nag_deviates_f_dist.

3 Description

The deviate, fp, associated with the lower tail probability, p, of the F-distribution with degrees of freedom ν1 and ν2 is defined as the solution to
P ( F fp : ν1 ,ν2) = p = ν 1 12 ν1 ν 2 12 ν2 Γ ( ν1 + ν2 2 ) Γ ( ν1 2 ) Γ ( ν2 2 ) 0 fp F 12 (ν1-2) (ν2+ν1F) -12 (ν1+ν2) dF ,  
where ν1,ν2>0; 0fp<.
The value of fp is computed by means of a transformation to a beta distribution, Pβ(Bβ:a,b):
P(Ff:ν1,ν2)=Pβ (Bν1f ν1f+ν2 :ν1/2,ν2/2)  
and using a call to g01fec.
For very large values of both ν1 and ν2, greater than 105, a normal approximation is used. If only one of ν1 or ν2 is greater than 105 then a χ2 approximation is used; see Abramowitz and Stegun (1972).

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: p double Input
On entry: p, the lower tail probability from the required F-distribution.
Constraint: 0.0p<1.0.
2: df1 double Input
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
3: df2 double Input
On entry: the degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
4: 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

If on exit fail.code= NE_PROBAB_CLOSE_TO_TAIL, NE_REAL_ARG_GE, NE_REAL_ARG_LE or NE_REAL_ARG_LT, then g01fdc returns 0.0.
On any of the error conditions listed below except fail.code= NE_SOL_NOT_CONV g01fdc returns 0.0.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
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.
The probability is too close to 0.0 or 1.0. The value of fp cannot be computed. This will only occur when the large sample approximations are used.
On entry, p=value.
Constraint: p<1.0.
On entry, df1=value and df2=value.
Constraint: df1>0.0 and df2>0.0.
On entry, p=value.
Constraint: p0.0.
The solution has failed to converge. However, the result should be a reasonable approximation. Alternatively, g01fec can be used with a suitable setting of the argument tol.

7 Accuracy

The result should be accurate to five significant digits.

8 Parallelism and Performance

g01fdc is not threaded in any implementation.

9 Further Comments

For higher accuracy g01fec 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 until the end of data is reached.

10.1 Program Text

Program Text (g01fdce.c)

10.2 Program Data

Program Data (g01fdce.d)

10.3 Program Results

Program Results (g01fdce.r)