NAG Library Routine Document
G01FFF returns the deviate associated with the given lower tail probability of the gamma distribution, via the routine name.
|REAL (KIND=nag_wp) G01FFF
||P, A, B, TOL
, associated with the lower tail probability,
, of the gamma distribution with shape parameter
and scale parameter
, 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
. The required
is found from the Taylor series expansion
is a starting approximation
For most values of
the starting value
is used, where
is the deviate associated with a lower tail probability of
for the standard Normal distribution.
close to zero,
is found to be a better starting value than
For small , is expressed in terms of an approximation to the exponential integral and 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.
Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the distribution Appl. Statist. 24 385–388
- 1: P – REAL (KIND=nag_wp)Input
On entry: , the lower tail probability from the required gamma distribution.
- 2: A – REAL (KIND=nag_wp)Input
On entry: , the shape parameter of the gamma distribution.
- 3: B – REAL (KIND=nag_wp)Input
On entry: , the scale parameter of the gamma distribution.
- 4: TOL – REAL (KIND=nag_wp)Input
: the relative accuracy required by you in the results. The smallest recommended value is
. If G01FFF is entered with TOL
or greater or equal to
is used instead.
- 5: IFAIL – INTEGERInput/Output
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
on exit, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, 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 , , or , then G01FFF returns .
is too close to
to enable the result to be calculated.
The solution has failed to converge in
iterations. A larger value of TOL
should be tried. The result may be a reasonable approximation.
The series to calculate the gamma function has failed to converge. This is an unlikely error exit.
In most cases the relative accuracy of the results should be as specified by TOL
. However, for very small values of
or very small values of
there may be some loss of accuracy.
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)