NAG Library Routine Document

1Purpose

g01sdf returns a number of lower or upper tail probabilities for the $F$ or variance-ratio distribution with real degrees of freedom.

2Specification

Fortran Interface
 Subroutine g01sdf ( tail, lf, f, ldf1, df1, ldf2, df2, p,
 Integer, Intent (In) :: ltail, lf, ldf1, ldf2 Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: f(lf), df1(ldf1), df2(ldf2) Real (Kind=nag_wp), Intent (Out) :: p(*) Character (1), Intent (In) :: tail(ltail)
#include nagmk26.h
 void g01sdf_ (const Integer *ltail, const char tail[], const Integer *lf, const double f[], const Integer *ldf1, const double df1[], const Integer *ldf2, const double df2[], double p[], Integer ivalid[], Integer *ifail, const Charlen length_tail)

3Description

The lower tail probability for the $F$, or variance-ratio, distribution with ${u}_{i}$ and ${v}_{i}$ degrees of freedom, $P\left({F}_{i}\le {f}_{i}:{u}_{i},{v}_{i}\right)$, is defined by:
 $P Fi ≤ fi :ui,vi = ui ui/2 vi vi/2 Γ ui + vi / 2 Γ ui/2 Γ vi/2 ∫ 0 fi Fi ui-2 / 2 ui Fi + vi - ui + vi / 2 d Fi ,$
for ${u}_{i}$, ${v}_{i}>0$, ${f}_{i}\ge 0$.
The probability is computed by means of a transformation to a beta distribution, ${P}_{{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$:
 $P Fi ≤ fi :ui,vi = Pβi Bi ≤ ui fi ui fi + vi : ui / 2 , vi / 2$
and using a call to g01eef.
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 whether the lower or upper tail probabilities are required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({F}_{i}\le {f}_{i}:{u}_{i},{v}_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({F}_{i}\ge {f}_{i}:{u}_{i},{v}_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:     $\mathbf{lf}$ – IntegerInput
On entry: the length of the array f.
Constraint: ${\mathbf{lf}}>0$.
4:     $\mathbf{f}\left({\mathbf{lf}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${f}_{i}$, the value of the $F$ variate with ${f}_{i}={\mathbf{f}}\left(j\right)$, .
Constraint: ${\mathbf{f}}\left(\mathit{j}\right)\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lf}}$.
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:     $\mathbf{p}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
On exit: ${p}_{i}$, the probabilities for the $F$-distribution.
10:   $\mathbf{ivalid}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
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 ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, ${f}_{i}<0.0$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${u}_{i}\le 0.0$, or ${v}_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=4$
The solution has failed to converge. The result returned should represent an approximation to the solution.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 arguments 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).

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: g01sdf 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 f, df1, df2 or tail 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: ${\mathbf{lf}}>0$.
${\mathbf{ifail}}=4$
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

g01sdf is not threaded in any implementation.

For higher accuracy g01sef 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 (g01sdfe.f90)

10.2Program Data

Program Data (g01sdfe.d)

10.3Program Results

Program Results (g01sdfe.r)

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