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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_inv_cdf_gamma_vector (g01tf)

## Purpose

nag_stat_inv_cdf_gamma_vector (g01tf) returns a number of deviates associated with given probabilities of the gamma distribution.

## Syntax

[g, ivalid, ifail] = g01tf(tail, p, a, b, 'ltail', ltail, 'lp', lp, 'la', la, 'lb', lb, 'tol', tol)
[g, ivalid, ifail] = nag_stat_inv_cdf_gamma_vector(tail, p, a, b, 'ltail', ltail, 'lp', lp, 'la', la, 'lb', lb, 'tol', tol)

## Description

The deviate, ${g}_{{p}_{i}}$, associated with the lower tail probability, ${p}_{i}$, of the gamma distribution with shape parameter ${\alpha }_{i}$ and scale parameter ${\beta }_{i}$, is defined as the solution to
 $P Gi ≤ gpi :αi,βi = pi = 1 βi αi Γ αi ∫ 0 gpi ei - Gi / βi Gi αi-1 dGi , 0 ≤ gpi < ∞ ; ​ αi , βi > 0 .$
The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the ${\chi }^{2}$-distribution.
Let ${y}_{i}=2\frac{{g}_{{p}_{i}}}{{\beta }_{i}}$. The required ${y}_{i}$ is found from the Taylor series expansion
 $yi=y0+∑rCry0 r! Eiϕy0 r,$
where ${y}_{0}$ is a starting approximation
• ${C}_{1}\left({u}_{i}\right)=1$,
• ${C}_{r+1}\left({u}_{i}\right)=\left(r\Psi +\frac{d}{d{u}_{i}}\right){C}_{r}\left({u}_{i}\right)$,
• ${\Psi }_{i}=\frac{1}{2}-\frac{{\alpha }_{i}-1}{{u}_{i}}$,
• ${E}_{i}={p}_{i}-\underset{0}{\overset{{y}_{0}}{\int }}{\varphi }_{i}\left({u}_{i}\right)d{u}_{i}$,
• ${\varphi }_{i}\left({u}_{i}\right)=\frac{1}{{2}^{{\alpha }_{i}}\Gamma \left({\alpha }_{i}\right)}{{e}_{i}}^{-{u}_{i}/2}{{u}_{i}}^{{\alpha }_{i}-1}$.
For most values of ${p}_{i}$ and ${\alpha }_{i}$ the starting value
 $y01=2αi zi⁢19αi +1-19αi 3$
is used, where ${z}_{i}$ is the deviate associated with a lower tail probability of ${p}_{i}$ for the standard Normal distribution.
For ${p}_{i}$ close to zero,
 $y02= piαi2αiΓ αi 1/αi$
is used.
For large ${p}_{i}$ values, when ${y}_{01}>4.4{\alpha }_{i}+6.0$,
 $y03=-2ln1-pi-αi-1ln12y01+lnΓ αi$
is found to be a better starting value than ${y}_{01}$.
For small ${\alpha }_{i}$ $\left({\alpha }_{i}\le 0.16\right)$, ${p}_{i}$ is expressed in terms of an approximation to the exponential integral and ${y}_{04}$ is found by Newton–Raphson iterations.
Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.
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

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ distribution Appl. Statist. 24 385–388

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{tail}\left({\mathbf{ltail}}\right)$ – cell array of strings
Indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., ${p}_{i}=P\left({G}_{i}\le {g}_{{p}_{i}}:{\alpha }_{i},{\beta }_{i}\right)$.
${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., ${p}_{i}=P\left({G}_{i}\ge {g}_{{p}_{i}}:{\alpha }_{i},{\beta }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left(\mathit{j}\right)=\text{'L'}$ or $\text{'U'}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
2:     $\mathrm{p}\left({\mathbf{lp}}\right)$ – double array
${p}_{i}$, the probability of the required gamma 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 .
3:     $\mathrm{a}\left({\mathbf{la}}\right)$ – double array
${\alpha }_{i}$, the first parameter of the required gamma distribution with ${\alpha }_{i}={\mathbf{a}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{a}}\left(\mathit{j}\right)\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
4:     $\mathrm{b}\left({\mathbf{lb}}\right)$ – double array
${\beta }_{i}$, the second parameter of the required gamma distribution with ${\beta }_{i}={\mathbf{b}}\left(j\right)$, .
Constraint: ${\mathbf{b}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.

### 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:     $\mathrm{lp}$int64int32nag_int scalar
Default: the dimension of the array p.
The length of the array p.
Constraint: ${\mathbf{lp}}>0$.
3:     $\mathrm{la}$int64int32nag_int scalar
Default: the dimension of the array a.
The length of the array a.
Constraint: ${\mathbf{la}}>0$.
4:     $\mathrm{lb}$int64int32nag_int scalar
Default: the dimension of the array b.
The length of the array b.
Constraint: ${\mathbf{lb}}>0$.
5:     $\mathrm{tol}$ – double scalar
Default: $0.0$
The relative accuracy required by you in the results. If nag_stat_inv_cdf_gamma_vector (g01tf) is entered with tol greater than or equal to $1.0$ or less than  (see nag_machine_precision (x02aj)), then the value of  is used instead.

### Output Parameters

1:     $\mathrm{g}\left(:\right)$ – double array
The dimension of the array g will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$
${g}_{{p}_{i}}$, the deviates for the gamma distribution.
2:     $\mathrm{ivalid}\left(:\right)$int64int32nag_int array
The dimension of the array ivalid will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$
${\mathbf{ivalid}}\left(i\right)$ 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 ${g}_{{p}_{i}}$.
${\mathbf{ivalid}}\left(i\right)=2$
 On entry, invalid value for ${p}_{i}$.
${\mathbf{ivalid}}\left(i\right)=3$
 On entry, ${\alpha }_{i}\le 0.0$, or ${\alpha }_{i}>{10}^{6}$, or ${\beta }_{i}\le 0.0$.
${\mathbf{ivalid}}\left(i\right)=4$
${p}_{i}$ is too close to $0.0$ or $1.0$ to enable the result to be calculated.
${\mathbf{ivalid}}\left(i\right)=5$
The solution has failed to converge. The result may be a reasonable approximation.
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_inv_cdf_gamma_vector (g01tf) 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 tail, p, a, or b was invalid.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
Constraint: ${\mathbf{la}}>0$.
${\mathbf{ifail}}=5$
Constraint: ${\mathbf{lb}}>0$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of ${\alpha }_{i}$ or very small values of ${p}_{i}$ there may be some loss of accuracy.

None.

## Example

This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.
```function g01tf_example

fprintf('g01tf example results\n\n');

tail = {'L'};
p = [0.01; 0.428; 0.869];
a = [1; 7.5; 45];
b = [20; 0.1; 10];

[x, ivalid, ifail] = g01tf( ...
tail, p, a, b);

fprintf('  tail  p       a       b         x     ivalid\n');
ltail = numel(tail);
lp    = numel(p);
la    = numel(a);
lb    = numel(b);
len  = max ([ltail, lp, la, lb]);
for i=0:len-1
fprintf('%5s%8.3f%8.3f%8.3f%10.3f%5d\n', tail{mod(i, ltail)+1}, ...
p(mod(i,lp)+1), a(mod(i,la)+1), b(mod(i,lb)+1), x(i+1), ivalid(i+1));
end

```
```g01tf example results

tail  p       a       b         x     ivalid
L   0.010   1.000  20.000     0.201    0
L   0.428   7.500   0.100     0.670    0
L   0.869  45.000  10.000   525.839    0
```