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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_inv_cdf_f_vector (g01td)

## Purpose

nag_stat_inv_cdf_f_vector (g01td) returns a number of deviates associated with given probabilities of the F$F$ or variance-ratio distribution with real degrees of freedom.

## Syntax

[f, ivalid, ifail] = g01td(tail, p, df1, df2, 'ltail', ltail, 'lp', lp, 'ldf1', ldf1, 'ldf2', ldf2)
[f, ivalid, ifail] = nag_stat_inv_cdf_f_vector(tail, p, df1, df2, 'ltail', ltail, 'lp', lp, 'ldf1', ldf1, 'ldf2', ldf2)

## Description

The deviate, fpi${f}_{{p}_{i}}$, associated with the lower tail probability, pi${p}_{i}$, of the F$F$-distribution with degrees of freedom ui${u}_{i}$ and vi${v}_{i}$ is defined as the solution to
 fpi P( Fi ≤ fpi : ui,vi) = pi = ( ui (1/2) ui vi (1/2) vi Γ (( ui + vi )/2) )/( Γ ((ui)/2) Γ ((vi)/2) ) ∫ Fi (1/2) (ui − 2) (vi + uiFi) − (1/2) (ui + vi) dFi, 0
$P( Fi ≤ fpi :ui,vi) = pi = u i 12 ui v i 12 vi Γ ( ui + vi 2 ) Γ ( ui 2 ) Γ ( vi 2 ) ∫ 0 fpi Fi 12 (ui-2) ( vi + ui Fi ) -12 ( ui + vi ) dFi ,$
where ui,vi > 0${u}_{i},{v}_{i}>0$; 0fpi < $0\le {f}_{{p}_{i}}<\infty$.
The value of fpi${f}_{{p}_{i}}$ is computed by means of a transformation to a beta distribution, Piβi( Bi βi : ai,bi) ${P}_{i{\beta }_{i}}\left({B}_{i}\le {\beta }_{i}:{a}_{i},{b}_{i}\right)$:
 P( Fi ≤ fpi : ui,vi) = Piβi( Bi ≤ ( ui fpi )/( ui fpi + vi ) : ui / 2 , vi / 2 ) $P( Fi ≤ fpi :ui,vi) = P iβi ( Bi ≤ ui fpi ui fpi + vi : ui / 2 , vi / 2 )$
and using a call to nag_stat_inv_cdf_beta_vector (g01te).
For very large values of both ui${u}_{i}$ and vi${v}_{i}$, greater than 105${10}^{5}$, a Normal approximation is used. If only one of ui${u}_{i}$ or vi${v}_{i}$ is greater than 105${10}^{5}$ then a χ2${\chi }^{2}$ 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 parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section [Vectorized s] 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:     tail(ltail) – cell array of strings
ltail, the dimension of the array, must satisfy the constraint ltail > 0${\mathbf{ltail}}>0$.
Indicates which tail the supplied probabilities represent. For j = ((i1)  mod  ltail) + 1 , for i = 1,2,,max (ltail,lp,ldf1,ldf2)$\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$:
tail(j) = 'L'${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., pi = P( Fi fpi : ui , vi ) ${p}_{i}=P\left({F}_{i}\le {f}_{{p}_{i}}:{u}_{i},{v}_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., pi = P( Fi fpi : ui , vi ) ${p}_{i}=P\left({F}_{i}\ge {f}_{{p}_{i}}:{u}_{i},{v}_{i}\right)$.
Constraint: tail(j) = 'L'${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or 'U'$\text{'U'}$, for j = 1,2,,ltail$\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
2:     p(lp) – double array
lp, the dimension of the array, must satisfy the constraint lp > 0${\mathbf{lp}}>0$.
pi${p}_{i}$, the probability of the required F$F$-distribution as defined by tail with pi = p(j)${p}_{i}={\mathbf{p}}\left(j\right)$, j = ((i1)  mod  lp) + 1.
Constraints:
• if tail(k) = 'L'${\mathbf{tail}}\left(k\right)=\text{'L'}$, 0.0p(j) < 1.0$0.0\le {\mathbf{p}}\left(\mathit{j}\right)<1.0$;
• otherwise 0.0 < p(j)1.0$0.0<{\mathbf{p}}\left(\mathit{j}\right)\le 1.0$.
Where k = (i1)  mod  ltail + 1 and j = (i1)  mod  lp + 1.
3:     df1(ldf1) – double array
ldf1, the dimension of the array, must satisfy the constraint ldf1 > 0${\mathbf{ldf1}}>0$.
ui${u}_{i}$, the degrees of freedom of the numerator variance with ui = df1(j)${u}_{i}={\mathbf{df1}}\left(j\right)$, j = ((i1)  mod  ldf1) + 1.
Constraint: df1(j) > 0.0${\mathbf{df1}}\left(\mathit{j}\right)>0.0$, for j = 1,2,,ldf1$\mathit{j}=1,2,\dots ,{\mathbf{ldf1}}$.
4:     df2(ldf2) – double array
ldf2, the dimension of the array, must satisfy the constraint ldf2 > 0${\mathbf{ldf2}}>0$.
vi${v}_{i}$, the degrees of freedom of the denominator variance with vi = df2(j)${v}_{i}={\mathbf{df2}}\left(j\right)$, j = ((i1)  mod  ldf2) + 1.
Constraint: df2(j) > 0.0${\mathbf{df2}}\left(\mathit{j}\right)>0.0$, for j = 1,2,,ldf2$\mathit{j}=1,2,\dots ,{\mathbf{ldf2}}$.

### Optional Input Parameters

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

None.

### Output Parameters

1:     f( : $:$) – double array
Note: the dimension of the array f must be at least max (ltail,lp,ldf1,ldf2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
fpi${f}_{{p}_{i}}$, the deviates for the F$F$-distribution.
2:     ivalid( : $:$) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lp,ldf1,ldf2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{ldf1}},{\mathbf{ldf2}}\right)$.
ivalid(i)${\mathbf{ivalid}}\left(i\right)$ indicates any errors with the input arguments, with
ivalid(i) = 0${\mathbf{ivalid}}\left(i\right)=0$
No error.
ivalid(i) = 1${\mathbf{ivalid}}\left(i\right)=1$
 On entry, invalid value supplied in tail when calculating fpi${f}_{{p}_{i}}$.
ivalid(i) = 2${\mathbf{ivalid}}\left(i\right)=2$
 On entry, invalid value for pi${p}_{i}$.
ivalid(i) = 3${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ui ≤ 0.0${u}_{i}\le 0.0$, or vi ≤ 0.0${v}_{i}\le 0.0$.
ivalid(i) = 4${\mathbf{ivalid}}\left(i\right)=4$
The solution has not converged. The result should still be a reasonable approximation to the solution.
ivalid(i) = 5${\mathbf{ivalid}}\left(i\right)=5$
The value of pi${p}_{i}$ is too close to 0.0$0.0$ or 1.0$1.0$ for the result to be computed. This will only occur when the large sample approximations are used.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Note: nag_stat_inv_cdf_f_vector (g01td) 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 ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of tail, p, df1, df2 was invalid, or the solution failed to converge.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ltail > 0${\mathbf{ltail}}>0$.
ifail = 3${\mathbf{ifail}}=3$
Constraint: lp > 0${\mathbf{lp}}>0$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: ldf1 > 0${\mathbf{ldf1}}>0$.
ifail = 5${\mathbf{ifail}}=5$
Constraint: ldf2 > 0${\mathbf{ldf2}}>0$.
ifail = 999${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The result should be accurate to five significant digits.

For higher accuracy nag_stat_inv_cdf_beta_vector (g01te) can be used along with the transformations given in Section [Description].

## Example

```function nag_stat_inv_cdf_f_vector_example
tail = {'L'};
p = [0.984; 0.9; 0.534];
df1 = [10; 1; 20.25];
df2 = [25.5; 1; 1];
[f, ivalid, ifail] = nag_stat_inv_cdf_f_vector(tail, p, df1, df2);

fprintf('\n   TAIL   P      DF1     DF2      F    IVALID\n');
ltail = numel(tail);
lp = numel(p);
ldf1 = numel(df1);
ldf2 = numel(df2);
len = max ([ltail, lp, ldf1, ldf2]);
for i=0:len-1
fprintf('%5c%7.3f%8.3f%8.3f%8.3f%7d\n', cell2mat(tail(mod(i, ltail)+1)), ...
p(mod(i,lp)+1), df1(mod(i,ldf1)+1), df2(mod(i,ldf2)+1), ...
f(i+1), ivalid(i+1));
end
```
```

TAIL   P      DF1     DF2      F    IVALID
L  0.984  10.000  25.500   2.847      0
L  0.900   1.000   1.000  39.863      0
L  0.534  20.250   1.000   2.498      0

```
```function g01td_example
tail = {'L'};
p = [0.984; 0.9; 0.534];
df1 = [10; 1; 20.25];
df2 = [25.5; 1; 1];
[f, ivalid, ifail] = g01td(tail, p, df1, df2);

fprintf('\n   TAIL   P      DF1     DF2      F    IVALID\n');
ltail = numel(tail);
lp = numel(p);
ldf1 = numel(df1);
ldf2 = numel(df2);
len = max ([ltail, lp, ldf1, ldf2]);
for i=0:len-1
fprintf('%5c%7.3f%8.3f%8.3f%8.3f%7d\n', cell2mat(tail(mod(i, ltail)+1)), ...
p(mod(i,lp)+1), df1(mod(i,ldf1)+1), df2(mod(i,ldf2)+1), ...
f(i+1), ivalid(i+1));
end
```
```

TAIL   P      DF1     DF2      F    IVALID
L  0.984  10.000  25.500   2.847      0
L  0.900   1.000   1.000  39.863      0
L  0.534  20.250   1.000   2.498      0

```