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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_prob_f_vector (g01sd)

## Purpose

nag_stat_prob_f_vector (g01sd) returns a number of lower or upper tail probabilities for the $F$ or variance-ratio distribution with real degrees of freedom.

## Syntax

[p, ivalid, ifail] = g01sd(tail, f, df1, df2, 'ltail', ltail, 'lf', lf, 'ldf1', ldf1, 'ldf2', ldf2)
[p, ivalid, ifail] = nag_stat_prob_f_vector(tail, f, df1, df2, 'ltail', ltail, 'lf', lf, 'ldf1', ldf1, 'ldf2', ldf2)

## Description

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 nag_stat_prob_beta (g01ee).
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 function 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 Vectorized Routines in the G01 Chapter Introduction for further information.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{tail}\left({\mathbf{ltail}}\right)$ – cell array of strings
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 $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
2:     $\mathrm{f}\left({\mathbf{lf}}\right)$ – double array
${f}_{i}$, 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 $\mathit{j}=1,2,\dots ,{\mathbf{lf}}$.
3:     $\mathrm{df1}\left({\mathbf{ldf1}}\right)$ – double array
${u}_{i}$, 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 $\mathit{j}=1,2,\dots ,{\mathbf{ldf1}}$.
4:     $\mathrm{df2}\left({\mathbf{ldf2}}\right)$ – double array
${v}_{i}$, 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 $\mathit{j}=1,2,\dots ,{\mathbf{ldf2}}$.

### Optional Input Parameters

1:     $\mathrm{ltail}$int64int32nag_int scalar
Default: the dimension of the array tail.
The length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:     $\mathrm{lf}$int64int32nag_int scalar
Default: the dimension of the array f.
The length of the array f.
Constraint: ${\mathbf{lf}}>0$.
3:     $\mathrm{ldf1}$int64int32nag_int scalar
Default: the dimension of the array df1.
The length of the array df1.
Constraint: ${\mathbf{ldf1}}>0$.
4:     $\mathrm{ldf2}$int64int32nag_int scalar
Default: the dimension of the array df2.
The length of the array df2.
Constraint: ${\mathbf{ldf2}}>0$.

### Output Parameters

1:     $\mathrm{p}\left(:\right)$ – double array
The dimension of the array p will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$
${p}_{i}$, the probabilities for the $F$-distribution.
2:     $\mathrm{ivalid}\left(:\right)$int64int32nag_int array
The dimension of the array ivalid will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lf}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$
${\mathbf{ivalid}}\left(i\right)$ 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.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_stat_prob_f_vector (g01sd) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W  ${\mathbf{ifail}}=1$
On entry, at least one value of f, df1, df2 or tail was invalid, or the solution failed to converge.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{lf}}>0$.
${\mathbf{ifail}}=4$
Constraint: ${\mathbf{ldf1}}>0$.
${\mathbf{ifail}}=5$
Constraint: ${\mathbf{ldf2}}>0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The result should be accurate to five significant digits.

For higher accuracy nag_stat_prob_beta_vector (g01se) can be used along with the transformations given in Description.

## Example

This example reads values from, and degrees of freedom for, a number of $F$-distributions and computes the associated lower tail probabilities.
```function g01sd_example

fprintf('g01sd example results\n\n');

f = [5.5; 39.9; 2.5];
df1 = [1.5; 1; 20.25];
df2 = [25.5; 1; 1];
tail = {'L'};
% calculate probability
[prob, ivalid, ifail] = g01sd( ...
tail, f, df1, df2);

fprintf('    F       df1    df2     prob\n');
lf    = numel(f);
ldf1  = numel(df1);
ldf2  = numel(df2);
ltail = numel(tail);
len   = max ([lf, ldf1, ldf2, ltail]);
for i=0:len-1
fprintf('%7.3f%8.3f%8.3f%8.3f\n', f(mod(i,lf)+1), df1(mod(i,ldf1)+1), ...
df2(mod(i,ldf2)+1), prob(i+1));
end

```
```g01sd example results

F       df1    df2     prob
5.500   1.500  25.500   0.984
39.900   1.000   1.000   0.900
2.500  20.250   1.000   0.534
```