NAG C Library Function Document

1Purpose

nag_deviates_gamma_vector (g01tfc) returns a number of deviates associated with given probabilities of the gamma distribution.

2Specification

 #include #include
 void nag_deviates_gamma_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer la, const double a[], Integer lb, const double b[], double tol, double g[], Integer ivalid[], NagError *fail)

3Description

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 Section 2.6 in the g01 Chapter Introduction for further information.

4References

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

5Arguments

1:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability, i.e., ${p}_{i}=P\left({G}_{i}\le {g}_{{p}_{i}}:{\alpha }_{i},{\beta }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability, i.e., ${p}_{i}=P\left({G}_{i}\ge {g}_{{p}_{i}}:{\alpha }_{i},{\beta }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lp}$IntegerInput
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4:    $\mathbf{p}\left[{\mathbf{lp}}\right]$const doubleInput
On entry: ${p}_{i}$, the probability of the required gamma distribution as defined by tail with ${p}_{i}={\mathbf{p}}\left[j\right]$, .
Constraints:
• if ${\mathbf{tail}}\left[k\right]=\mathrm{Nag_LowerTail}$, $0.0\le {\mathbf{p}}\left[\mathit{j}\right]<1.0$;
• otherwise $0.0<{\mathbf{p}}\left[\mathit{j}\right]\le 1.0$.
Where  and .
5:    $\mathbf{la}$IntegerInput
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6:    $\mathbf{a}\left[{\mathbf{la}}\right]$const doubleInput
On entry: ${\alpha }_{i}$, the first parameter of the required gamma distribution with ${\alpha }_{i}={\mathbf{a}}\left[j\right]$, .
Constraint: $0.0<{\mathbf{a}}\left[\mathit{j}-1\right]\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7:    $\mathbf{lb}$IntegerInput
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8:    $\mathbf{b}\left[{\mathbf{lb}}\right]$const doubleInput
On entry: ${\beta }_{i}$, the second parameter of the required gamma distribution with ${\beta }_{i}={\mathbf{b}}\left[j\right]$, .
Constraint: ${\mathbf{b}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9:    $\mathbf{tol}$doubleInput
On entry: the relative accuracy required by you in the results. If nag_deviates_gamma_vector (g01tfc) is entered with tol greater than or equal to $1.0$ or less than  (see nag_machine_precision (X02AJC)), the value of  is used instead.
10:  $\mathbf{g}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array g must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${g}_{{p}_{i}}$, the deviates for the gamma distribution.
11:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${g}_{{p}_{i}}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, invalid value for ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${\alpha }_{i}\le 0.0$, or ${\alpha }_{i}>{10}^{6}$, or ${\beta }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
${p}_{i}$ is too close to $0.0$ or $1.0$ to enable the result to be calculated.
${\mathbf{ivalid}}\left[i-1\right]=5$
The solution has failed to converge. The result may be a reasonable approximation.
12:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lp}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID
On entry, at least one value of tail, p, a, or b was invalid.

7Accuracy

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.

8Parallelism and Performance

nag_deviates_gamma_vector (g01tfc) is not threaded in any implementation.

None.

10Example

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

10.1Program Text

Program Text (g01tfce.c)

10.2Program Data

Program Data (g01tfce.d)

10.3Program Results

Program Results (g01tfce.r)