# NAG Library Routine Document

## 1Purpose

g01gdf returns the probability associated with the lower tail of the noncentral $F$ or variance-ratio distribution.

## 2Specification

Fortran Interface
 Function g01gdf ( f, df1, df2, tol,
 Real (Kind=nag_wp) :: g01gdf Integer, Intent (In) :: maxit Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: f, df1, df2, rlamda, tol
#include <nagmk26.h>
 double g01gdf_ (const double *f, const double *df1, const double *df2, const double *rlamda, const double *tol, const Integer *maxit, Integer *ifail)

## 3Description

The lower tail probability of the noncentral $F$-distribution with ${\nu }_{1}$ and ${\nu }_{2}$ degrees of freedom and noncentrality parameter $\lambda$, $P\left(F\le f:{\nu }_{1},{\nu }_{2}\text{;}\lambda \right)$, is defined by
 $PF≤f:ν1,ν2;λ=∫0xpF:ν1,ν2;λdF,$
where
 $PF : ν1,ν2;λ =∑j= 0∞e-λ/2 λ/2jj! ×ν1+ 2jν1+ 2j/2 ν2ν2/2 Bν1+ 2j/2,ν2/2$
 $×uν1+2j-2/2ν2+ν1+2ju -ν1+2j+ν2/2$
and $B\left(·,·\right)$ is the beta function.
The probability is computed by means of a transformation to a noncentral beta distribution:
 $PF≤f:ν1,ν2;λ=PβX≤x:a,b;λ,$
where $x=\frac{{\nu }_{1}f}{{\nu }_{1}f+{\nu }_{2}}$ and ${P}_{\beta }\left(X\le x:a,b\text{;}\lambda \right)$ is the lower tail probability integral of the noncentral beta distribution with parameters $a$, $b$, and $\lambda$.
If ${\nu }_{2}$ is very large, greater than ${10}^{6}$, then a ${\chi }^{2}$ approximation is used.

## 4References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

## 5Arguments

1:     $\mathbf{f}$ – Real (Kind=nag_wp)Input
On entry: $f$, the deviate from the noncentral $F$-distribution.
Constraint: ${\mathbf{f}}>0.0$.
2:     $\mathbf{df1}$ – Real (Kind=nag_wp)Input
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: $0.0<{\mathbf{df1}}\le {10}^{6}$.
3:     $\mathbf{df2}$ – Real (Kind=nag_wp)Input
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{df2}}>0.0$.
4:     $\mathbf{rlamda}$ – Real (Kind=nag_wp)Input
On entry: $\lambda$, the noncentrality parameter.
Constraint: $0.0\le {\mathbf{rlamda}}\le -2.0\mathrm{log}\left(U\right)$ where $U$ is the safe range parameter as defined by x02amf.
5:     $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: the relative accuracy required by you in the results. If g01gdf is entered with tol greater than or equal to $1.0$ or less than  (see x02ajf), the value of  is used instead.
6:     $\mathbf{maxit}$ – IntegerInput
On entry: the maximum number of iterations to be used.
Suggested value: $500$. See g01gcf and g01gef for further details.
Constraint: ${\mathbf{maxit}}\ge 1$.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g01gdf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$ or ${\mathbf{3}}$, then g01gdf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{df1}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{df1}}\le {10}^{6}$.
On entry, ${\mathbf{df1}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df1}}>0.0$.
On entry, ${\mathbf{df2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df2}}>0.0$.
On entry, ${\mathbf{f}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{f}}>0.0$.
On entry, ${\mathbf{maxit}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxit}}\ge 1$.
On entry, ${\mathbf{rlamda}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0\le {\mathbf{rlamda}}\le -2.0×\mathrm{log}\left(U\right)$, where $U$ is the safe range parameter as defined by x02amf.
${\mathbf{ifail}}=2$
The solution has failed to converge in $〈\mathit{\text{value}}〉$ iterations. Consider increasing maxit or tol.
${\mathbf{ifail}}=3$
The required probability cannot be computed accurately. This may happen if the result would be very close to zero or one. Alternatively the values of df1 and f may be too large. In the latter case you could try using a normal approximation, see Abramowitz and Stegun (1972).
${\mathbf{ifail}}=4$
The required accuracy was not achieved when calculating the initial value of the central $F$ or ${\chi }^{2}$ probability. You should try a larger value of tol. If the ${\chi }^{2}$ approximation is being used then g01gdf returns zero otherwise the value returned should be an approximation to the correct value.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The relative accuracy should be as specified by tol. For further details see g01gcf and g01gef.

## 8Parallelism and Performance

g01gdf is not threaded in any implementation.

When both ${\nu }_{1}$ and ${\nu }_{2}$ are large a Normal approximation may be used and when only ${\nu }_{1}$ is large a ${\chi }^{2}$ approximation may be used. In both cases $\lambda$ is required to be of the same order as ${\nu }_{1}$. See Abramowitz and Stegun (1972) for further details.

## 10Example

This example reads values from, and degrees of freedom for, $F$-distributions, computes the lower tail probabilities and prints all these values until the end of data is reached.

### 10.1Program Text

Program Text (g01gdfe.f90)

### 10.2Program Data

Program Data (g01gdfe.d)

### 10.3Program Results

Program Results (g01gdfe.r)