nag_prob_f_dist (g01edc) (PDF version)
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g01 Chapter Introduction
NAG C 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.
tail=Nag_LowerTail
The lower tail probability is returned, i.e., PFf:ν1,ν2.
tail=Nag_UpperTail
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.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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.
NE_PROBAB_CLOSE_TO_TAIL
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.
NE_REAL_ARG_LE
On entry, df1=value, df2=value.
Constraint: df1>0.0 and df2>0.0.
NE_REAL_ARG_LT
On entry, f=value.
Constraint: f0.0.

7  Accuracy

The result should be accurate to five significant digits.

8  Further Comments

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

9  Example

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

9.1  Program Text

Program Text (g01edce.c)

9.2  Program Data

Program Data (g01edce.d)

9.3  Program Results

Program Results (g01edce.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012