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 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_prob_gamma_vector (g01sfc) is not threaded in any implementation.

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 Document

nag_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

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