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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, gpi${g}_{{p}_{i}}$, associated with the lower tail probability, pi${p}_{i}$, of the gamma distribution with shape parameter αi${\alpha }_{i}$ and scale parameter βi${\beta }_{i}$, is defined as the solution to
 gpi P( Gi ≤ gpi : αi,βi) = pi = 1/( βiαi Γ (αi) ) ∫ ei − Gi / βi Giαi − 1dGi,  0 ≤ gpi < ∞; ​αi,βi > 0. 0
$P( Gi ≤ gpi :αi,βi) = pi = 1 βi αi Γ (αi) ∫ 0 gpi ei - Gi / βi Gi αi-1 dGi , 0 ≤ gpi < ∞ ; ​ αi , βi > 0 .$
The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the χ2${\chi }^{2}$-distribution.
Let yi = 2(gpi)/(βi) ${y}_{i}=2\frac{{g}_{{p}_{i}}}{{\beta }_{i}}$. The required yi${y}_{i}$ is found from the Taylor series expansion
 yi = y0 + ∑r(Cr(y0))/(r ! ) ((Ei)/(φ(y0)))r, $yi=y0+∑rCr(y0) r! (Eiϕ(y0) ) r,$
where y0${y}_{0}$ is a starting approximation
• C1(ui) = 1${C}_{1}\left({u}_{i}\right)=1$,
• Cr + 1(ui) = (rΨ + d/(dui)) Cr(ui)${C}_{r+1}\left({u}_{i}\right)=\left(r\Psi +\frac{d}{d{u}_{i}}\right){C}_{r}\left({u}_{i}\right)$,
• Ψi = (1/2)(αi1)/(ui) ${\Psi }_{i}=\frac{1}{2}-\frac{{\alpha }_{i}-1}{{u}_{i}}$,
• Ei = pi0y0φi(ui)dui${E}_{i}={p}_{i}-\underset{0}{\overset{{y}_{0}}{\int }}{\varphi }_{i}\left({u}_{i}\right)d{u}_{i}$,
• φi(ui) = 1/(2αiΓ (αi) )eiui / 2uiαi1${\varphi }_{i}\left({u}_{i}\right)=\frac{1}{{2}^{{\alpha }_{i}}\Gamma \left({\alpha }_{i}\right)}{{e}_{i}}^{-{u}_{i}/2}{{u}_{i}}^{{\alpha }_{i}-1}$.
For most values of pi${p}_{i}$ and αi${\alpha }_{i}$ the starting value
 y01 = 2αi (zi×sqrt(1/(9αi)) + 1 − 1/(9αi))3 $y01=2αi (zi⁢19αi +1-19αi ) 3$
is used, where zi${z}_{i}$ is the deviate associated with a lower tail probability of pi${p}_{i}$ for the standard Normal distribution.
For pi${p}_{i}$ close to zero,
 y02 = (piαi2αiΓ(αi))1 / αi $y02= (piαi2αiΓ (αi) ) 1/αi$
is used.
For large pi${p}_{i}$ values, when y01 > 4.4αi + 6.0${y}_{01}>4.4{\alpha }_{i}+6.0$,
 y03 = − 2[ln(1 − pi) − (αi − 1)ln((1/2)y01) + ln(Γ(αi))] $y03=-2[ln(1-pi)-(αi-1)ln(12y01)+ln(Γ (αi) ) ]$
is found to be a better starting value than y01${y}_{01}$.
For small αi${\alpha }_{i}$ (αi0.16)$\left({\alpha }_{i}\le 0.16\right)$, pi${p}_{i}$ is expressed in terms of an approximation to the exponential integral and y04${y}_{04}$ 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 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

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2${\chi }^{2}$ distribution Appl. Statist. 24 385–388

## 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,la,lb)$\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$:
tail(j) = 'L'${\mathbf{tail}}\left(j\right)=\text{'L'}$
The lower tail probability, i.e., pi = P( Gi gpi : αi , βi ) ${p}_{i}=P\left({G}_{i}\le {g}_{{p}_{i}}:{\alpha }_{i},{\beta }_{i}\right)$.
tail(j) = 'U'${\mathbf{tail}}\left(j\right)=\text{'U'}$
The upper tail probability, i.e., pi = P( Gi gpi : αi , βi ) ${p}_{i}=P\left({G}_{i}\ge {g}_{{p}_{i}}:{\alpha }_{i},{\beta }_{i}\right)$.
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 gamma 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:     a(la) – double array
la, the dimension of the array, must satisfy the constraint la > 0${\mathbf{la}}>0$.
αi${\alpha }_{i}$, the first parameter of the required gamma distribution with αi = a(j)${\alpha }_{i}={\mathbf{a}}\left(j\right)$, j = ((i1)  mod  la) + 1.
Constraint: 0.0 < a(j)106$0.0<{\mathbf{a}}\left(\mathit{j}\right)\le {10}^{6}$, for j = 1,2,,la$\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
4:     b(lb) – double array
lb, the dimension of the array, must satisfy the constraint lb > 0${\mathbf{lb}}>0$.
βi${\beta }_{i}$, the second parameter of the required gamma distribution with βi = b(j)${\beta }_{i}={\mathbf{b}}\left(j\right)$, j = ((i1)  mod  lb) + 1.
Constraint: b(j) > 0.0${\mathbf{b}}\left(\mathit{j}\right)>0.0$, for j = 1,2,,lb$\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.

### 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:     la – int64int32nag_int scalar
Default: The dimension of the array a.
The length of the array a.
Constraint: la > 0${\mathbf{la}}>0$.
4:     lb – int64int32nag_int scalar
Default: The dimension of the array b.
The length of the array b.
Constraint: lb > 0${\mathbf{lb}}>0$.
5:     tol – double scalar
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$1.0$ or less than 10 × machine precision (see nag_machine_precision (x02aj)), then the value of 10 × machine precision is used instead.
Default: 0.0$0.0$

None.

### Output Parameters

1:     g( : $:$) – double array
Note: the dimension of the array g must be at least max (ltail,lp,la,lb)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$.
gpi${g}_{{p}_{i}}$, the deviates for the gamma distribution.
2:     ivalid( : $:$) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lp,la,lb)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$.
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 gpi${g}_{{p}_{i}}$.
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, αi ≤ 0.0${\alpha }_{i}\le 0.0$, or αi > 106${\alpha }_{i}>{10}^{6}$, or βi ≤ 0.0${\beta }_{i}\le 0.0$.
ivalid(i) = 4${\mathbf{ivalid}}\left(i\right)=4$
pi${p}_{i}$ is too close to 0.0$0.0$ or 1.0$1.0$ to enable the result to be calculated.
ivalid(i) = 5${\mathbf{ivalid}}\left(i\right)=5$
The solution has failed to converge. The result may be a reasonable approximation.
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_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 ifail = 1${\mathbf{ifail}}=1$
On entry, at least one value of tail, p, a, or b was invalid.
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: la > 0${\mathbf{la}}>0$.
ifail = 5${\mathbf{ifail}}=5$
Constraint: lb > 0${\mathbf{lb}}>0$.
ifail = 999${\mathbf{ifail}}=-999$
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 αi${\alpha }_{i}$ or very small values of pi${p}_{i}$ there may be some loss of accuracy.

None.

## Example

```function nag_stat_inv_cdf_gamma_vector_example
tail = {'L'};
p = [0.01; 0.428; 0.869];
a = [1; 7.5; 45];
b = [20; 0.1; 10];
[x, ivalid, ifail] = nag_stat_inv_cdf_gamma_vector(tail, p, a, b);

fprintf('\n  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('%5c%8.3f%8.3f%8.3f%10.3f%5d\n', cell2mat(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
```
```

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

```
```function g01tf_example
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('\n  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('%5c%8.3f%8.3f%8.3f%10.3f%5d\n', cell2mat(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
```
```

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

```