# NAG Library Routine Document

## 1Purpose

g01tdf returns a number of deviates associated with given probabilities of the $F$ or variance-ratio distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Subroutine g01tdf ( tail, lp, p, ldf1, df1, ldf2, df2, f,
 Integer, Intent (In) :: ltail, lp, ldf1, ldf2 Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: p(lp), df1(ldf1), df2(ldf2) Real (Kind=nag_wp), Intent (Out) :: f(*) Character (1), Intent (In) :: tail(ltail)
#include <nagmk26.h>
 void g01tdf_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *ldf1, const double df1[], const Integer *ldf2, const double df2[], double f[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

## 3Description

The deviate, ${f}_{{p}_{i}}$, associated with the lower tail probability, ${p}_{i}$, of the $F$-distribution with degrees of freedom ${u}_{i}$ and ${v}_{i}$ is defined as the solution to
 $P Fi ≤ fpi :ui,vi = pi = u i 12 ui v i 12 vi Γ ui + vi 2 Γ ui 2 Γ vi 2 ∫ 0 fpi Fi 12 ui-2 vi + ui Fi -12 ui + vi dFi ,$
where ${u}_{i},{v}_{i}>0$; $0\le {f}_{{p}_{i}}<\infty$.
The value of ${f}_{{p}_{i}}$ is computed by means of a transformation to a beta distribution, ${P}_{i{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$:
 $P Fi ≤ fpi :ui,vi = P iβi Bi ≤ ui fpi ui fpi + vi : ui / 2 , vi / 2$
and using a call to g01tef.
For very large values of both ${u}_{i}$ and ${v}_{i}$, greater than ${10}^{5}$, a Normal approximation is used. If only one of ${u}_{i}$ or ${v}_{i}$ is greater than ${10}^{5}$ then a ${\chi }^{2}$ approximation is used; see Abramowitz and Stegun (1972).
The input arrays to this routine are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the G01 Chapter Introduction for further information.

## 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:     $\mathbf{ltail}$ – IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     $\mathbf{tail}\left({\mathbf{ltail}}\right)$ – Character(1) arrayInput
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., ${p}_{i}=P\left({F}_{i}\le {f}_{{p}_{i}}:{u}_{i},{v}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., ${p}_{i}=P\left({F}_{i}\ge {f}_{{p}_{i}}:{u}_{i},{v}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:     $\mathbf{lp}$ – IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:     $\mathbf{p}\left({\mathbf{lp}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{i}$, the probability of the required $F$-distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left(j\right)$, .
Constraints:
• if ${\mathbf{tail}}\left(k\right)=\text{'L'}$, $0.0\le {\mathbf{p}}\left(\mathit{j}\right)<1.0$;
• otherwise $0.0<{\mathbf{p}}\left(\mathit{j}\right)\le 1.0$.
Where  and .
5:     $\mathbf{ldf1}$ – IntegerInput
On entry: the length of the array df1.
Constraint: ${\mathbf{ldf1}}>0$.
6:     $\mathbf{df1}\left({\mathbf{ldf1}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${u}_{i}$, the degrees of freedom of the numerator variance with ${u}_{i}={\mathbf{df1}}\left(j\right)$, .
Constraint: ${\mathbf{df1}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf1}}$.
7:     $\mathbf{ldf2}$ – IntegerInput
On entry: the length of the array df2.
Constraint: ${\mathbf{ldf2}}>0$.
8:     $\mathbf{df2}\left({\mathbf{ldf2}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${v}_{i}$, the degrees of freedom of the denominator variance with ${v}_{i}={\mathbf{df2}}\left(j\right)$, .
Constraint: ${\mathbf{df2}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{ldf2}}$.
9:     $\mathbf{f}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array f must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
On exit: ${f}_{{p}_{i}}$, the deviates for the $F$-distribution.
10:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
On exit: ${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating ${f}_{{p}_{i}}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, invalid value for ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${u}_{i}\le 0.0$, or ${v}_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=4$
The solution has not converged. The result should still be a reasonable approximation to the solution.
${\mathbf{ivalid}}\left(i\right)=5$
The value of ${p}_{i}$ is too close to $0.0$ or $1.0$ for the result to be computed. This will only occur when the large sample approximations are used.
11:   $\mathbf{ifail}$ – 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: g01tdf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, at least one value of tail, p, df1, df2 was invalid, or the solution failed to converge.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf1}}>0$.
${\mathbf{ifail}}=5$
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldf2}}>0$.
${\mathbf{ifail}}=-99$
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 result should be accurate to five significant digits.

## 8Parallelism and Performance

g01tdf is not threaded in any implementation.

For higher accuracy g01tef can be used along with the transformations given in Section 3.

## 10Example

This example reads the lower tail probabilities for several $F$-distributions, and calculates and prints the corresponding deviates.

### 10.1Program Text

Program Text (g01tdfe.f90)

### 10.2Program Data

Program Data (g01tdfe.d)

### 10.3Program Results

Program Results (g01tdfe.r)