nag_prob_f_dist (g01edc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_prob_f_dist (g01edc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_prob_f_dist (g01edc) returns the probability for the lower or upper tail of the F or variance-ratio distribution with real degrees of freedom.

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_prob_f_dist (Nag_TailProbability tail, double f, double df1, double df2, NagError *fail)

3  Description

The lower tail probability for the F, or variance-ratio distribution, with ν1 and ν2 degrees of freedom, PFf:ν1,ν2, is defined by:
PFf:ν1,ν2=ν1ν1/2ν2ν2/2 Γ ν1+ν2/2 Γν1/2 Γν2/2 0fFν1-2/2ν1F+ν2- ν1+ν2/2dF,
for ν1, ν2>0, f0.
The probability 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_prob_beta_dist (g01eec).
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:     tailNag_TailProbabilityInput
On entry: indicates whether an upper or lower tail probability is required.
The lower tail probability is returned, i.e., PFf:ν1,ν2.
The upper tail probability is returned, i.e., PFf:ν1,ν2.
Constraint: tail=Nag_LowerTail or Nag_UpperTail.
2:     fdoubleInput
On entry: f, the value of the F variate.
Constraint: f0.0.
3:     df1doubleInput
On entry: the degrees of freedom of the numerator variance, ν1.
Constraint: df1>0.0.
4:     df2doubleInput
On entry: the degrees of freedom of the denominator variance, ν2.
Constraint: df2>0.0.
5:     failNagError *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 NE_PROBAB_CLOSE_TO_TAIL nag_prob_f_dist (g01edc) returns 0.0.
Dynamic memory allocation failed.
On entry, argument value had an illegal value.
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.
The probability is too close to 0.0 or 1.0. f is too far out into the tails for the probability to be evaluated exactly. The result tends to approach 1.0 if f is large, or 0.0 if f is small. The result returned is a good approximation to the required solution.
On entry, df1=value and df2=value.
Constraint: df1>0.0 and df2>0.0.
On entry, f=value.
Constraint: f0.0.

7  Accuracy

The result should be accurate to five significant digits.

8  Parallelism and Performance

Not applicable.

9  Further Comments

For higher accuracy nag_prob_beta_dist (g01eec) can be used along with the transformations given in Section 3.

10  Example

This example reads values from, and degrees of freedom for, a number of F-distributions and computes the associated lower tail probabilities.

10.1  Program Text

Program Text (g01edce.c)

10.2  Program Data

Program Data (g01edce.d)

10.3  Program Results

Program Results (g01edce.r)

nag_prob_f_dist (g01edc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014