g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_prob_f_dist (g01edc)

## 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 #include
 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 ${\nu }_{1}$ and ${\nu }_{2}$ degrees of freedom, $P\left(F\le f:{\nu }_{1},{\nu }_{2}\right)$, is defined by:
 $PF≤f:ν1,ν2=ν1ν1/2ν2ν2/2 Γ ν1+ν2/2 Γν1/2 Γν2/2 ∫0fFν1-2/2ν1F+ν2- ν1+ν2/2dF,$
for ${\nu }_{1}$, ${\nu }_{2}>0$, $f\ge 0$.
The probability is computed by means of a transformation to a beta distribution, ${P}_{\beta }\left(B\le \beta :a,b\right)$:
 $PF≤f:ν1,ν2=Pβ B≤ν1f ν1f+ν2 :ν1/2,ν2/2$
and using a call to nag_prob_beta_dist (g01eec).
For very large values of both ${\nu }_{1}$ and ${\nu }_{2}$, greater than ${10}^{5}$, a normal approximation is used. If only one of ${\nu }_{1}$ or ${\nu }_{2}$ is greater than ${10}^{5}$ then a ${\chi }^{2}$ 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.
${\mathbf{tail}}=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., $P\left(F\le f:{\nu }_{1},{\nu }_{2}\right)$.
${\mathbf{tail}}=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., $P\left(F\ge f:{\nu }_{1},{\nu }_{2}\right)$.
Constraint: ${\mathbf{tail}}=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$.
2:     fdoubleInput
On entry: $f$, the value of the $F$ variate.
Constraint: ${\mathbf{f}}\ge 0.0$.
3:     df1doubleInput
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{df1}}>0.0$.
4:     df2doubleInput
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
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.
On entry, argument $〈\mathit{\text{value}}〉$ 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, ${\mathbf{df1}}=〈\mathit{\text{value}}〉$, ${\mathbf{df2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df1}}>0.0$ and ${\mathbf{df2}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{f}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{f}}\ge 0.0$.

## 7  Accuracy

The result should be accurate to five significant digits.

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)