NAG FL Interfaceg01tff (inv_​cdf_​gamma_​vector)

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1Purpose

g01tff returns a number of deviates associated with given probabilities of the gamma distribution.

2Specification

Fortran Interface
 Subroutine g01tff ( tail, lp, p, la, a, lb, b, tol, g,
 Integer, Intent (In) :: ltail, lp, la, lb Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(*) Real (Kind=nag_wp), Intent (In) :: p(lp), a(la), b(lb), tol Real (Kind=nag_wp), Intent (Out) :: g(*) Character (1), Intent (In) :: tail(ltail)
C Header Interface
#include <nag.h>
 void g01tff_ (const Integer *ltail, const char tail[], const Integer *lp, const double p[], const Integer *la, const double a[], const Integer *lb, const double b[], const double *tol, double g[], Integer ivalid[], Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01tff or nagf_stat_inv_cdf_gamma_vector.

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
 $yi=y0+∑rCr(y0) r! (Eiϕ(y0) ) r,$
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 ${y}_{01}>4.4{\alpha }_{i}+6.0$,
 $y03=−2⁢[ln(1-pi)-(αi-1)ln(12y01)+ln(Γ(αi))]$
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 routine 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: $\mathbf{ltail}$Integer Input
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2: $\mathbf{tail}\left({\mathbf{ltail}}\right)$Character(1) array Input
On entry: indicates which tail the supplied probabilities represent. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lp}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left(j\right)=\text{'L'}$
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 $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3: $\mathbf{lp}$Integer Input
On entry: the length of the array p.
Constraint: ${\mathbf{lp}}>0$.
4: $\mathbf{p}\left({\mathbf{lp}}\right)$Real (Kind=nag_wp) array Input
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 ${\mathbf{tail}}\left(k\right)=\text{'L'}$, $0.0\le {\mathbf{p}}\left(\mathit{j}\right)<1.0$;
• otherwise $0.0<{\mathbf{p}}\left(\mathit{j}\right)\le 1.0$.
Where and .
5: $\mathbf{la}$Integer Input
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6: $\mathbf{a}\left({\mathbf{la}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\alpha }_{i}$, the first parameter of the required gamma distribution with ${\alpha }_{i}={\mathbf{a}}\left(j\right)$, .
Constraint: $0.0<{\mathbf{a}}\left(\mathit{j}\right)\le {10}^{6}$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7: $\mathbf{lb}$Integer Input
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8: $\mathbf{b}\left({\mathbf{lb}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\beta }_{i}$, the second parameter of the required gamma distribution with ${\beta }_{i}={\mathbf{b}}\left(j\right)$, .
Constraint: ${\mathbf{b}}\left(\mathit{j}\right)>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9: $\mathbf{tol}$Real (Kind=nag_wp) Input
On entry: the relative accuracy required by you in the results. If g01tff is entered with tol greater than or equal to $1.0$ or less than (see x02ajf), the value of is used instead.
10: $\mathbf{g}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension 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: $\mathbf{ivalid}\left(*\right)$Integer array Output
Note: the dimension 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: ${\mathbf{ivalid}}\left(i\right)$ 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.
12: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g01tff may return useful information.
${\mathbf{ifail}}=1$
On entry, at least one value of tail, p, a, or b was invalid.
Check ivalid for more information.
${\mathbf{ifail}}=2$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ltail}}>0$.
${\mathbf{ifail}}=3$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lp}}>0$.
${\mathbf{ifail}}=4$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{la}}>0$.
${\mathbf{ifail}}=5$
On entry, $\text{array size}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lb}}>0$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

Background information to multithreading can be found in the Multithreading documentation.
g01tff 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 (g01tffe.f90)

10.2Program Data

Program Data (g01tffe.d)

10.3Program Results

Program Results (g01tffe.r)