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G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01EDF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G01EDF returns the probability for the lower or upper tail of the $F$ or variance-ratio distribution with real degrees of freedom, via the routine name.

## 2  Specification

 FUNCTION G01EDF ( TAIL, F, DF1, DF2, IFAIL)
 REAL (KIND=nag_wp) G01EDF
 INTEGER IFAIL REAL (KIND=nag_wp) F, DF1, DF2 CHARACTER(1) TAIL

## 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 G01EEF.
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  Parameters

1:     $\mathrm{TAIL}$ – CHARACTER(1)Input
On entry: indicates whether an upper or lower tail probability is required.
${\mathbf{TAIL}}=\text{'L'}$
The lower tail probability is returned, i.e., $P\left(F\le f:{\nu }_{1},{\nu }_{2}\right)$.
${\mathbf{TAIL}}=\text{'U'}$
The upper tail probability is returned, i.e., $P\left(F\ge f:{\nu }_{1},{\nu }_{2}\right)$.
Constraint: ${\mathbf{TAIL}}=\text{'L'}$ or $\text{'U'}$.
2:     $\mathrm{F}$ – REAL (KIND=nag_wp)Input
On entry: $f$, the value of the $F$ variate.
Constraint: ${\mathbf{F}}\ge 0.0$.
3:     $\mathrm{DF1}$ – REAL (KIND=nag_wp)Input
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{DF1}}>0.0$.
4:     $\mathrm{DF2}$ – REAL (KIND=nag_wp)Input
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{DF2}}>0.0$.
5:     $\mathrm{IFAIL}$ – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ 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 parameters may be useful even if ${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: G01EDF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If ${\mathbf{IFAIL}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{3}}$ on exit, then G01EDF returns $0.0$.
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{TAIL}}\ne \text{'L'}$ or $\text{'U'}$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{F}}<0.0$.
${\mathbf{IFAIL}}=3$
 On entry, ${\mathbf{DF1}}\le 0.0$, or ${\mathbf{DF2}}\le 0.0$.
${\mathbf{IFAIL}}=4$
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.
${\mathbf{IFAIL}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

## 7  Accuracy

The result should be accurate to five significant digits.

Not applicable.

## 9  Further Comments

For higher accuracy G01EEF 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 (g01edfe.f90)

### 10.2  Program Data

Program Data (g01edfe.d)

### 10.3  Program Results

Program Results (g01edfe.r)