# NAG Library Function Document

## 1Purpose

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.

## 2Specification

 #include #include
 double nag_prob_f_dist (Nag_TailProbability tail, double f, double df1, double df2, NagError *fail)

## 3Description

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).

## 4References

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

## 5Arguments

1:    $\mathbf{tail}$Nag_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:    $\mathbf{f}$doubleInput
On entry: $f$, the value of the $F$ variate.
Constraint: ${\mathbf{f}}\ge 0.0$.
3:    $\mathbf{df1}$doubleInput
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{df1}}>0.0$.
4:    $\mathbf{df2}$doubleInput
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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, ${\mathbf{df1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{df2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df1}}>0.0$ and ${\mathbf{df2}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{f}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{f}}\ge 0.0$.

## 7Accuracy

The result should be accurate to five significant digits.

## 8Parallelism and Performance

nag_prob_f_dist (g01edc) is not threaded in any implementation.

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

## 10Example

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

### 10.1Program Text

Program Text (g01edce.c)

### 10.2Program Data

Program Data (g01edce.d)

### 10.3Program Results

Program Results (g01edce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017