void	g01sfc (Integer ltail, const Nag_TailProbability tail[], Integer lg, const double g[], Integer la, const double a[], Integer lb, const double b[], double p[], Integer ivalid[], NagError *fail)

The function may be called by the names: g01sfc, nag_stat_prob_gamma_vector or nag_prob_gamma_vector.

3 Description

The lower tail probability for the gamma distribution with parameters

α_{i}

and

β_{i}

P (G_{i} \leq g_{i})

, is defined by:

P (G_{i} \leq g_{i} : α_{i}, β_{i}) = \frac{1}{{β_{i}}^{α_{i}} Γ (α_{i})} \int_{0}^{g_{i}} {G_{i}}^{α_{i} - 1} e^{- G_{i} / β_{i}} d G_{i}, α_{i} > 0.0, ​ β_{i} > 0.0 .

The mean of the distribution is

α_{i} β_{i}

and its variance is

α_{i} {β_{i}}^{2}

. The transformation

Z_{i} = \frac{G_{i}}{β_{i}}

is applied to yield the following incomplete gamma function in normalized form,

P (G_{i} \leq g_{i} : α_{i}, β_{i}) = P (Z_{i} \leq g_{i} / β_{i} : α_{i}, 1.0) = \frac{1}{Γ (α_{i})} \int_{0}^{g_{i} / β_{i}} {Z_{i}}^{α_{i} - 1} e^{- Z_{i}} d Z_{i} .

This is then evaluated using s14bac.

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

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: $ltail$ – Integer Input

On entry: the length of the array tail.

Constraint:

ltail > 0

2: $tail [ltail]$ – const Nag_TailProbability Input

On entry: indicates whether a lower or upper tail probability is required. For

j = (i - 1) mod ltail

, for

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

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

Constraint:

tail [j - 1] = Nag_LowerTail

Nag_UpperTail

, for

j = 1, 2, \dots, ltail

3: $\lg$ – Integer Input

On entry: the length of the array g.

Constraint:

\lg > 0

4: $g [\lg]$ – const double Input

On entry:

g_{i}

, the value of the gamma variate with

g_{i} = g [j]

j = (i - 1) mod \lg

Constraint:

g [j - 1] \geq 0.0

, for

j = 1, 2, \dots, \lg

5: $la$ – Integer Input

On entry: the length of the array a.

Constraint:

la > 0

6: $a [la]$ – const double Input

On entry: the parameter

α_{i}

of the gamma distribution with

α_{i} = a [j]

j = (i - 1) mod la

Constraint:

a [j - 1] > 0.0

, for

j = 1, 2, \dots, la

7: $lb$ – Integer Input

On entry: the length of the array b.

Constraint:

lb > 0

8: $b [lb]$ – const double Input

On entry: the parameter

β_{i}

of the 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: $p [\dim]$ – double Output

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

\max (\lg, la, lb, ltail)

On exit:

p_{i}

, the probabilities of the beta distribution.

10: $ivalid [\dim]$ – Integer Output

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

\max (\lg, la, lb, ltail)

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 $p_{i}$ .
$ivalid [i - 1] = 2$: On entry, $g_{i} < 0.0$ .
$ivalid [i - 1] = 3$: On entry, $α_{i} \leq 0.0$ , or, $β_{i} \leq 0.0$ .
$ivalid [i - 1] = 4$: The solution did not converge in $600$ iterations, see s14bac for details. The probability returned should be a reasonable approximation to the solution.

11: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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: $\lg > 0$ .

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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID: On entry, at least one value of g, a, b or tail was invalid, or the solution did not converge.
Check ivalid for more information.

7 Accuracy

The result should have a relative accuracy of machine precision. There are rare occasions when the relative accuracy attained is somewhat less than machine precision but the error should not exceed more than

1

2

decimal places.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g01sfc is not threaded in any implementation.

9 Further Comments

The time taken by g01sfc to calculate each probability varies slightly with the input arguments

g_{i}

α_{i}

and

β_{i}

10 Example

This example reads in values from a number of gamma distributions and computes the associated lower tail probabilities.

g01sf: FL CL CPP AD PY MB

NAG CL Interfaceg01sfc (prob_​gamma_​vector)

▸▿ 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 CL Interface
g01sfc (prob_gamma_vector)