nag_deviates_f_dist (g01fdc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_deviates_f_dist (g01fdc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_deviates_f_dist (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>
#include <nagg01.h>
double  nag_deviates_f_dist (double p, double df1, double df2, NagError *fail)

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 + ν1 F -12 ν1 + ν2 dF ,  
where ν1,ν2>0; 0fp<.
The value of fp is computed by means of a transformation to a beta distribution, PβBβ:a,b:
PFf:ν1,ν2=Pβ Bν1f ν1f+ν2 :ν1/2,ν2/2  
and using a call to nag_deviates_beta (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 doubleInput
On entry: p, the lower tail probability from the required F-distribution.
Constraint: 0.0p<1.0.
2:     df1 doubleInput
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
3:     df2 doubleInput
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 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On any of the error conditions listed below except fail.code= NE_SOL_NOT_CONV nag_deviates_f_dist (g01fdc) returns 0.0.
Dynamic memory allocation failed.
See Section in the Essential Introduction 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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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, nag_deviates_beta (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

Not applicable.

9  Further Comments

For higher accuracy nag_deviates_beta (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)

nag_deviates_f_dist (g01fdc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015