NAG Library Routine Document
G01FFF
1 Purpose
G01FFF returns the deviate associated with the given lower tail probability of the gamma distribution, via the routine name.
2 Specification
REAL (KIND=nag_wp) G01FFF |
INTEGER |
IFAIL |
REAL (KIND=nag_wp) |
P, A, B, TOL |
|
3 Description
The deviate,
gp, associated with the lower tail probability,
p, of the gamma distribution with shape parameter
α and scale parameter
β, is defined as the solution to
The method used is described by
Best and Roberts (1975) making use of the relationship between the gamma distribution and the
χ2-distribution.
Let
y=2
gpβ
. The required
y is found from the Taylor series expansion
where
y0 is a starting approximation
- C1u=1,
- Cr+1u=rΨ+
ddu
Cru,
- Ψ=12-
α-1u
,
- E=p-∫0y0ϕudu,
- ϕu=
12αΓ
α
e-u/2uα-1.
For most values of
p and
α the starting value
is used, where
z is the deviate associated with a lower tail probability of
p for the standard Normal distribution.
For
p close to zero,
is used.
For large
p values, when
y01>4.4α+6.0,
is found to be a better starting value than
y01.
For small α α≤0.16, p is expressed in terms of an approximation to the exponential integral and y04 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.
4 References
Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the
χ2 distribution
Appl. Statist. 24 385–388
5 Parameters
- 1: P – REAL (KIND=nag_wp)Input
On entry: p, the lower tail probability from the required gamma distribution.
Constraint:
0.0≤P<1.0.
- 2: A – REAL (KIND=nag_wp)Input
On entry: α, the shape parameter of the gamma distribution.
Constraint:
0.0<A≤106.
- 3: B – REAL (KIND=nag_wp)Input
On entry: β, the scale parameter of the gamma distribution.
Constraint:
B>0.0.
- 4: TOL – REAL (KIND=nag_wp)Input
On entry: the relative accuracy required by you in the results. The smallest recommended value is
50×δ, where
δ=max10-18,machine precision. If G01FFF is entered with
TOL less than
50×δ or greater or equal to
1.0, then
50×δ is used instead.
- 5: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
IFAIL≠0 on exit, the recommended value is
-1.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
On exit:
IFAIL=0 unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: G01FFF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit IFAIL=1, 2, 3 or 5, then G01FFF returns 0.0.
- IFAIL=1
- IFAIL=2
On entry, | A≤0.0, |
or | A>106, |
or | B≤0.0 |
- IFAIL=3
P is too close to
0.0 or
1.0 to enable the result to be calculated.
- IFAIL=4
The solution has failed to converge in
100 iterations. A larger value of
TOL should be tried. The result may be a reasonable approximation.
- IFAIL=5
The series to calculate the gamma function has failed to converge. This is an unlikely error exit.
7 Accuracy
In most cases the relative accuracy of the results should be as specified by
TOL. However, for very small values of
α or very small values of
p there may be some loss of accuracy.
8 Further Comments
None.
9 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.
9.1 Program Text
Program Text (g01fffe.f90)
9.2 Program Data
Program Data (g01fffe.d)
9.3 Program Results
Program Results (g01fffe.r)