hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_gamma_vector (g01tf)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_inv_cdf_gamma_vector (g01tf) returns a number of deviates associated with given probabilities of the gamma distribution.


[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)


The deviate, gpi, associated with the lower tail probability, pi, of the gamma distribution with shape parameter αi and scale parameter βi, 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 χ2-distribution.
Let yi=2 gpiβi . The required yi is found from the Taylor series expansion
yi=y0+rCry0 r! Eiϕy0 r,  
where y0 is a starting approximation
For most values of pi and αi the starting value
y01=2αi zi19αi +1-19αi 3  
is used, where zi is the deviate associated with a lower tail probability of pi for the standard Normal distribution.
For pi close to zero,
y02= piαi2αiΓ αi 1/αi  
is used.
For large pi values, when y01>4.4αi+6.0,
y03=-2ln1-pi-αi-1ln12y01+lnΓ αi  
is found to be a better starting value than y01.
For small αi αi0.16, pi 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.
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 Vectorized Routines in the G01 Chapter Introduction for further information.


Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2 distribution Appl. Statist. 24 385–388


Compulsory Input Parameters

1:     tailltail – cell array of strings
Indicates which tail the supplied probabilities represent. For j= i-1 mod ltail +1 , for i=1,2,,maxltail,lp,la,lb:
The lower tail probability, i.e., pi = P Gi gpi : αi , βi .
The upper tail probability, i.e., pi = P Gi gpi : αi , βi .
Constraint: tailj='L' or 'U', for j=1,2,,ltail.
2:     plp – double array
pi, the probability of the required gamma distribution as defined by tail with pi=pj, j=i-1 mod lp+1.
  • if tailk='L', 0.0pj<1.0;
  • otherwise 0.0<pj1.0.
Where k=i-1 mod ltail+1 and j=i-1 mod lp+1.
3:     ala – double array
αi, the first parameter of the required gamma distribution with αi=aj, j=i-1 mod la+1.
Constraint: 0.0<aj106, for j=1,2,,la.
4:     blb – double array
βi, the second parameter of the required gamma distribution with βi=bj, j=i-1 mod lb+1.
Constraint: bj>0.0, for j=1,2,,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.
2:     lp int64int32nag_int scalar
Default: the dimension of the array p.
The length of the array p.
Constraint: lp>0.
3:     la int64int32nag_int scalar
Default: the dimension of the array a.
The length of the array a.
Constraint: la>0.
4:     lb int64int32nag_int scalar
Default: the dimension of the array b.
The length of the array b.
Constraint: lb>0.
5:     tol – double scalar
Default: 0.0
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 or less than 10×machine precision (see nag_machine_precision (x02aj)), then the value of 10×machine precision is used instead.

Output Parameters

1:     g: – double array
The dimension of the array g will be maxltail,lp,la,lb
gpi, the deviates for the gamma distribution.
2:     ivalid: int64int32nag_int array
The dimension of the array ivalid will be maxltail,lp,la,lb
ivalidi indicates any errors with the input arguments, with
No error.
On entry,invalid value supplied in tail when calculating gpi.
On entry,invalid value for pi.
On entry,αi0.0,
pi is too close to 0.0 or 1.0 to enable the result to be calculated.
The solution has failed to converge. The result may be a reasonable approximation.
3:     ifail int64int32nag_int scalar
ifail=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
On entry, at least one value of tail, p, a, or b was invalid.
Check ivalid for more information.
Constraint: ltail>0.
Constraint: lp>0.
Constraint: la>0.
Constraint: lb>0.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of αi or very small values of pi there may be some loss of accuracy.

Further Comments



This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.
function g01tf_example

fprintf('g01tf example results\n\n');

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('  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('%5s%8.3f%8.3f%8.3f%10.3f%5d\n', 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));

g01tf example results

  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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015