$C_{1} (u_{i}) = 1$ ,
$C_{r + 1} (u_{i}) = (r Ψ + \frac{d}{d u_{i}}) C_{r} (u_{i})$ ,
$Ψ_{i} = \frac{1}{2} - \frac{α_{i} - 1}{u_{i}}$ ,
$E_{i} = p_{i} - \int_{0}^{y_{0}} ϕ_{i} (u_{i}) d u_{i}$ ,
$ϕ_{i} (u_{i}) = \frac{1}{2^{α_{i}} Γ (α_{i})} {e_{i}}^{- u_{i} / 2} {u_{i}}^{α_{i} - 1}$ .

For most values of

p_{i}

and

α_{i}

the starting value

y_{01} = 2 α_{i} {(z_{i} \sqrt{\frac{1}{9 α_{i}}} + 1 - \frac{1}{9 α_{i}})}^{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,

y_{02} = {(p_{i} α_{i} 2^{α_{i}} Γ (α_{i}))}^{1 / α_{i}}

is used.

For large

p_{i}

values, when

y_{01} > 4.4 α_{i} + 6.0

y_{03} = - 2 [\ln (1 - p_{i}) - (α_{i} - 1) \ln (\frac{1}{2} y_{01}) + \ln (Γ (α_{i}))]

is found to be a better starting value than

y_{01}

For small

α_{i}

(α_{i} \leq 0.16)

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 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.

References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the

χ^{2}

distribution Appl. Statist. 24 385–388

Parameters

Compulsory Input Parameters

1: $tail (ltail)$ – cell array of strings

Indicates which tail the supplied probabilities represent. For

j = ((i - 1) mod ltail) + 1

, for

i = 1, 2, \dots, \max (ltail, lp, la, lb)

$tail (j) ='L'$: The lower tail probability, i.e., $p_{i} = P (G_{i} \leq g_{p_{i}} : α_{i}, β_{i})$ .
$tail (j) ='U'$: The upper tail probability, i.e., $p_{i} = P (G_{i} \geq g_{p_{i}} : α_{i}, β_{i})$ .

Constraint:

tail (j) ='L'

'U'

, for

j = 1, 2, \dots, ltail

2: $p (lp)$ – double array

p_{i}

, the probability of the required gamma distribution as defined by tail with

p_{i} = p (j)

j = ((i - 1) mod lp) + 1

Constraints:

if $tail (k) ='L'$ , $0.0 \leq p (j) < 1.0$ ;
otherwise $0.0 < p (j) \leq 1.0$ .

Where

k = (i - 1) mod ltail + 1

and

j = (i - 1) mod lp + 1

3: $a (la)$ – double array

α_{i}

, the first parameter of the required gamma distribution with

α_{i} = a (j)

j = ((i - 1) mod la) + 1

Constraint:

0.0 < a (j) \leq 10^{6}

, for

j = 1, 2, \dots, la

4: $b (lb)$ – double array

β_{i}

, the second parameter of the required gamma distribution with

β_{i} = b (j)

j = ((i - 1) mod lb) + 1

Constraint:

b (j) > 0.0

, for

j = 1, 2, \dots, 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 $10 \times machine precision$ (see nag_machine_precision (x02aj)), then the value of $10 \times machine precision$ is used instead.

Output Parameters

1: $g (:)$ – double array

The dimension of the array g will be

\max (ltail, lp, la, lb)

g_{p_{i}}

, the deviates for the gamma distribution.

2: $ivalid (:)$ – int64int32nag_int array

The dimension of the array ivalid will be

\max (ltail, lp, la, lb)

ivalid (i)

indicates any errors with the input arguments, with

$ivalid (i) = 0$

No error.

$ivalid (i) = 1$

On entry,

invalid value supplied in tail when calculating

g_{p_{i}}

$ivalid (i) = 2$

On entry,

invalid value for

p_{i}

$ivalid (i) = 3$

On entry,	$α_{i} \leq 0.0$ ,
or	$α_{i} > 10^{6}$ ,
or	$β_{i} \leq 0.0$ .

$ivalid (i) = 4$

p_{i}

is too close to

0.0

1.0

to enable the result to be calculated.

$ivalid (i) = 5$

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.

$ifail = 2$: Constraint: $ltail > 0$ .

$ifail = 3$: Constraint: $lp > 0$ .

$ifail = 4$: Constraint: $la > 0$ .

$ifail = 5$: Constraint: $lb > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: 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

α_{i}

or very small values of

p_{i}

there may be some loss of accuracy.

Further Comments

None.

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.

Open in the MATLAB editor: g01tf_example

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));
end

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