NAG Library Function Document

nag_deviates_gamma_vector (g01tfc)

void	nag_deviates_gamma_vector (Integer ltail, const Nag_TailProbability tail[], Integer lp, const double p[], Integer la, const double a[], Integer lb, const double b[], double tol, double g[], Integer ivalid[], NagError *fail)

3 Description

The deviate,

g_{p_{i}}

, associated with the lower tail probability,

p_{i}

, of the gamma distribution with shape parameter

α_{i}

and scale parameter

β_{i}

, is defined as the solution to

P (G_{i} \leq g_{p_{i}} : α_{i}, β_{i}) = p_{i} = \frac{1}{{β_{i}}^{α_{i}} Γ (α_{i})} \int_{0}^{g_{p_{i}}} {e_{i}}^{- G_{i} / β_{i}} {G_{i}}^{α_{i} - 1} d G_{i}, 0 \leq g_{p_{i}} < \infty; ​ α_{i}, β_{i} > 0 .

The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the

χ^{2}

-distribution.

Let

y_{i} = 2 \frac{g_{p_{i}}}{β_{i}}

. The required

y_{i}

is found from the Taylor series expansion

y_{i} = y_{0} + \sum_{r} \frac{C_{r} (y_{0})}{r!} {(\frac{E_{i}}{ϕ (y_{0})})}^{r},

where

y_{0}

is a starting approximation

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

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 Section 2.6 in the g01 Chapter Introduction for further information.

4 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

5 Arguments

1: ltail – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: tail[ltail] – const Nag_TailProbabilityInput

On entry: indicates which tail the supplied probabilities represent. For

j = (i - 1) mod ltail

, for

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

$tail [j] = Nag_LowerTail$: The lower tail probability, i.e., $p_{i} = P (G_{i} \leq g_{p_{i}} : α_{i}, β_{i})$ .
$tail [j] = Nag_UpperTail$: The upper tail probability, i.e., $p_{i} = P (G_{i} \geq g_{p_{i}} : α_{i}, β_{i})$ .

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: lp – IntegerInput

On entry: the length of the array p.

Constraint:

lp > 0

4: p[lp] – const doubleInput

On entry:

p_{i}

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

p_{i} = p [j]

j = (i - 1) mod lp

Constraints:

if $tail [k] = Nag_LowerTail$ , $0.0 \leq p [j] < 1.0$ ;
otherwise $0.0 < p [j] \leq 1.0$ .

Where

k = (i - 1) mod ltail

and

j = (i - 1) mod lp

5: la – IntegerInput

On entry: the length of the array a.

Constraint:

la > 0

6: a[la] – const doubleInput

On entry:

α_{i}

, the first parameter of the required gamma distribution with

α_{i} = a [j]

j = (i - 1) mod la

Constraint:

0.0 < a [j - 1] \leq 10^{6}

, for

j = 1, 2, \dots, la

7: lb – IntegerInput

On entry: the length of the array b.

Constraint:

lb > 0

8: b[lb] – const doubleInput

On entry:

β_{i}

, the second parameter of the required gamma distribution with

β_{i} = b [j]

j = (i - 1) mod lb

Constraint:

b [j - 1] > 0.0

, for

j = 1, 2, \dots, lb

9: tol – doubleInput

On entry: the relative accuracy required by you in the results. If nag_deviates_gamma_vector (g01tfc) is entered with tol greater than or equal to

1.0

or less than

10 \times machine precision

(see nag_machine_precision (X02AJC)), then the value of

10 \times machine precision

is used instead.

10: g[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array g must be at least

\max (ltail, lp, la, lb)

On exit:

g_{p_{i}}

, the deviates for the gamma distribution.

11: ivalid[ $\dim$ ] – IntegerOutput

Note: the dimension, dim, of the array ivalid must be at least

\max (ltail, lp, la, lb)

On exit:

ivalid [i - 1]

indicates any errors with the input arguments, with

$ivalid [i - 1] = 0$

No error.

$ivalid [i - 1] = 1$

On entry,

invalid value supplied in tail when calculating

g_{p_{i}}

$ivalid [i - 1] = 2$

On entry,

invalid value for

p_{i}

$ivalid [i - 1] = 3$

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

$ivalid [i - 1] = 4$

p_{i}

is too close to

0.0

1.0

to enable the result to be calculated.

$ivalid [i - 1] = 5$

The solution has failed to converge. The result may be a reasonable approximation.

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_SIZE: On entry, $array size = ⟨value⟩$ .
Constraint: $la > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $lb > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $lp > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $ltail > 0$ .
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NW_IVALID: On entry, at least one value of tail, p, a, or b was invalid.
Check ivalid for more information.

7 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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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.

NAG Library Function Documentnag_deviates_gamma_vector (g01tfc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_deviates_gamma_vector (g01tfc)