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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_gamma_vector (g01tf)

Purpose

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

Syntax

[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)

Description

The deviate, gpigpi, associated with the lower tail probability, pipi, of the gamma distribution with shape parameter αiαi and scale parameter βiβi, is defined as the solution to
gpi
P( Gi gpi : αi,βi) = pi = 1/( βiαi Γ (αi) )ei Gi / βi Giαi1dGi,  0gpi < ; ​αi,βi > 0.
0
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 χ2χ2-distribution.
Let yi = 2(gpi)/(βi) yi=2 gpiβi . The required yiyi is found from the Taylor series expansion
yi = y0 + r(Cr(y0))/(r ! ) ((Ei)/(φ(y0)))r,
yi=y0+rCr(y0) r! (Eiϕ(y0) ) r,
where y0y0 is a starting approximation
For most values of pipi and αiαi the starting value
y01 = 2αi (zi×sqrt(1/(9αi)) + 11/(9αi))3
y01=2αi (zi19αi +1-19αi ) 3
is used, where zizi is the deviate associated with a lower tail probability of pipi for the standard Normal distribution.
For pipi close to zero,
y02 = (piαi2αiΓ(αi))1 / αi
y02= (piαi2αiΓ (αi) ) 1/αi
is used.
For large pipi values, when y01 > 4.4αi + 6.0y01>4.4αi+6.0,
y03 = 2[ln(1pi)(αi1)ln((1/2)y01) + ln(Γ(αi))]
y03=-2[ln(1-pi)-(αi-1)ln(12y01)+ln(Γ (αi) ) ]
is found to be a better starting value than y01y01.
For small αiαi (αi0.16)(αi0.16), pipi is expressed in terms of an approximation to the exponential integral and y04y04 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 parameters by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section [Vectorized s] in the G01 Chapter Introduction for further information.

References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the χ2χ2 distribution Appl. Statist. 24 385–388

Parameters

Compulsory Input Parameters

1:     tail(ltail) – cell array of strings
ltail, the dimension of the array, must satisfy the constraint ltail > 0ltail>0.
Indicates which tail the supplied probabilities represent. For j = ((i1)  mod  ltail) + 1 j= ( (i-1) mod ltail ) +1 , for i = 1,2,,max (ltail,lp,la,lb)i=1,2,,max(ltail,lp,la,lb):
tail(j) = 'L'tailj='L'
The lower tail probability, i.e., pi = P( Gi gpi : αi , βi ) pi = P( Gi gpi : αi , βi ) .
tail(j) = 'U'tailj='U'
The upper tail probability, i.e., pi = P( Gi gpi : αi , βi ) pi = P( Gi gpi : αi , βi ) .
Constraint: tail(j) = 'L'tailj='L' or 'U''U', for j = 1,2,,ltailj=1,2,,ltail.
2:     p(lp) – double array
lp, the dimension of the array, must satisfy the constraint lp > 0lp>0.
pipi, the probability of the required gamma distribution as defined by tail with pi = p(j)pi=pj, j = ((i1)  mod  lp) + 1j=((i-1) mod lp)+1.
Constraints:
  • if tail(k) = 'L'tailk='L', 0.0p(j) < 1.00.0pj<1.0;
  • otherwise 0.0 < p(j)1.00.0<pj1.0.
Where k = (i1)  mod  ltail + 1k=(i-1) mod ltail+1 and j = (i1)  mod  lp + 1j=(i-1) mod lp+1.
3:     a(la) – double array
la, the dimension of the array, must satisfy the constraint la > 0la>0.
αiαi, the first parameter of the required gamma distribution with αi = a(j)αi=aj, j = ((i1)  mod  la) + 1j=((i-1) mod la)+1.
Constraint: 0.0 < a(j)1060.0<aj106, for j = 1,2,,laj=1,2,,la.
4:     b(lb) – double array
lb, the dimension of the array, must satisfy the constraint lb > 0lb>0.
βiβi, the second parameter of the required gamma distribution with βi = b(j)βi=bj, j = ((i1)  mod  lb) + 1j=((i-1) mod lb)+1.
Constraint: b(j) > 0.0bj>0.0, for j = 1,2,,lbj=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 > 0ltail>0.
2:     lp – int64int32nag_int scalar
Default: The dimension of the array p.
The length of the array p.
Constraint: lp > 0lp>0.
3:     la – int64int32nag_int scalar
Default: The dimension of the array a.
The length of the array a.
Constraint: la > 0la>0.
4:     lb – int64int32nag_int scalar
Default: The dimension of the array b.
The length of the array b.
Constraint: lb > 0lb>0.
5:     tol – double scalar
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 1.01.0 or less than 10 × machine precision10×machine precision (see nag_machine_precision (x02aj)), then the value of 10 × machine precision10×machine precision is used instead.
Default: 0.00.0

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     g( : :) – double array
Note: the dimension of the array g must be at least max (ltail,lp,la,lb)max(ltail,lp,la,lb).
gpigpi, the deviates for the gamma distribution.
2:     ivalid( : :) – int64int32nag_int array
Note: the dimension of the array ivalid must be at least max (ltail,lp,la,lb)max(ltail,lp,la,lb).
ivalid(i)ivalidi indicates any errors with the input arguments, with
ivalid(i) = 0ivalidi=0
No error.
ivalid(i) = 1ivalidi=1
On entry,invalid value supplied in tail when calculating gpigpi.
ivalid(i) = 2ivalidi=2
On entry,invalid value for pipi.
ivalid(i) = 3ivalidi=3
On entry,αi0.0αi0.0,
orαi > 106αi>106,
orβi0.0βi0.0.
ivalid(i) = 4ivalidi=4
pipi is too close to 0.00.0 or 1.01.0 to enable the result to be calculated.
ivalid(i) = 5ivalidi=5
The solution has failed to converge. The result may be a reasonable approximation.
3:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1
On entry, at least one value of tail, p, a, or b was invalid.
Check ivalid for more information.
  ifail = 2ifail=2
Constraint: ltail > 0ltail>0.
  ifail = 3ifail=3
Constraint: lp > 0lp>0.
  ifail = 4ifail=4
Constraint: la > 0la>0.
  ifail = 5ifail=5
Constraint: lb > 0lb>0.
  ifail = 999ifail=-999
Dynamic memory allocation failed.

Accuracy

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

Further Comments

None.

Example

function nag_stat_inv_cdf_gamma_vector_example
tail = {'L'};
p = [0.01; 0.428; 0.869];
a = [1; 7.5; 45];
b = [20; 0.1; 10];
[x, ivalid, ifail] = nag_stat_inv_cdf_gamma_vector(tail, p, a, b);

fprintf('\n  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('%5c%8.3f%8.3f%8.3f%10.3f%5d\n', cell2mat(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));
end
 

  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

function g01tf_example
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('\n  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('%5c%8.3f%8.3f%8.3f%10.3f%5d\n', cell2mat(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));
end
 

  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


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