G01 Chapter Contents
G01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG01FDF

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

G01FDF returns the deviate associated with the given lower tail probability of the $F$ or variance-ratio distribution with real degrees of freedom, via the routine name.

## 2  Specification

 FUNCTION G01FDF ( P, DF1, DF2, IFAIL)
 REAL (KIND=nag_wp) G01FDF
 INTEGER IFAIL REAL (KIND=nag_wp) P, DF1, DF2

## 3  Description

The deviate, ${f}_{p}$, associated with the lower tail probability, $p$, of the $F$-distribution with degrees of freedom ${\nu }_{1}$ and ${\nu }_{2}$ is defined as the solution to
 $P F ≤ fp : ν1 ,ν2 = p = ν 1 12 ν1 ν 2 12 ν2 Γ ν1 + ν2 2 Γ ν1 2 Γ ν2 2 ∫ 0 fp F 12 ν1-2 ν2 + ν1 F -12 ν1 + ν2 dF ,$
where ${\nu }_{1},{\nu }_{2}>0$; $0\le {f}_{p}<\infty$.
The value of ${f}_{p}$ 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 G01FEF.
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:     P – REAL (KIND=nag_wp)Input
On entry: $p$, the lower tail probability from the required $F$-distribution.
Constraint: $0.0\le {\mathbf{P}}<1.0$.
2:     DF1 – REAL (KIND=nag_wp)Input
On entry: the degrees of freedom of the numerator variance, ${\nu }_{1}$.
Constraint: ${\mathbf{DF1}}>0.0$.
3:     DF2 – REAL (KIND=nag_wp)Input
On entry: the degrees of freedom of the denominator variance, ${\nu }_{2}$.
Constraint: ${\mathbf{DF2}}>0.0$.
4:     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: G01FDF 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 ${\mathbf{IFAIL}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{4}}$, then G01FDF returns $0.0$.
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{P}}<0.0$, or ${\mathbf{P}}\ge 1.0$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{DF1}}\le 0.0$, or ${\mathbf{DF2}}\le 0.0$.
${\mathbf{IFAIL}}=3$
The solution has not converged. The result should still be a reasonable approximation to the solution. Alternatively, G01FEF can be used with a suitable setting of the parameter TOL.
${\mathbf{IFAIL}}=4$
The value of P is too close to $0$ or $1$ for the value of ${f}_{p}$ to be computed. This will only occur when the large sample approximations are used.

## 7  Accuracy

The result should be accurate to five significant digits.

For higher accuracy G01FEF can be used along with the transformations given in Section 3.

## 9  Example

This example reads the lower tail probabilities for several $F$-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

### 9.1  Program Text

Program Text (g01fdfe.f90)

### 9.2  Program Data

Program Data (g01fdfe.d)

### 9.3  Program Results

Program Results (g01fdfe.r)