NAG Library Function Document

nag_prob_gamma_vector (g01sfc)

void	nag_prob_gamma_vector (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)

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 nag_incomplete_gamma (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 – IntegerInput

On entry: the length of the array tail.

Constraint:

ltail > 0

2: tail[ltail] – const Nag_TailProbabilityInput

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 – IntegerInput

On entry: the length of the array g.

Constraint:

\lg > 0

4: g[lg] – const doubleInput

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 – IntegerInput

On entry: the length of the array a.

Constraint:

la > 0

6: a[la] – const doubleInput

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 – IntegerInput

On entry: the length of the array b.

Constraint:

lb > 0

8: b[lb] – const doubleInput

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[ $\dim$ ] – doubleOutput

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[ $\dim$ ] – IntegerOutput

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 nag_incomplete_gamma (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 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: $\lg > 0$ .
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.
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

Not applicable.

9 Further Comments

The time taken by nag_prob_gamma_vector (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.

NAG Library Function Documentnag_prob_gamma_vector (g01sfc)

+− 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_prob_gamma_vector (g01sfc)