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NAG Library Function Document

nag_gamma_dist (g01efc)

▸▿ 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

nag_gamma_dist (g01efc) returns the lower or upper tail probability of the gamma distribution, with parameters

α

and

β

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_gamma_dist (Nag_TailProbability tail, double g, double a, double b, NagError *fail)

3 Description

The lower tail probability for the gamma distribution with parameters

α

and

β

P (G \leq g)

, is defined by:

P (G \leq g; α, β) = \frac{1}{β^{α} Γ (α)} \int_{0}^{g} G^{α - 1} e^{- G / β} d G, α > 0.0, ​ β > 0.0 .

The mean of the distribution is

α β

and its variance is

α β^{2}

. The transformation

Z = \frac{G}{β}

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

P (G \leq g; α, β) = P (Z \leq g / β : α, 1.0) = \frac{1}{Γ (α)} \int_{0}^{g / β} Z^{α - 1} e^{- Z} d Z .

This is then evaluated using nag_incomplete_gamma (s14bac).

4 References

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

5 Arguments

1: $tail$ – Nag_TailProbabilityInput

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

$tail = Nag_LowerTail$: The lower tail probability is returned, that is $P (G \leq g : α, β)$ .
$tail = Nag_UpperTail$: The upper tail probability is returned, that is $P (G \geq g : α, β)$ .

Constraint:

tail = Nag_LowerTail

Nag_UpperTail

2: $g$ – doubleInput

On entry:

g

, the value of the gamma variate.

Constraint:

g \geq 0.0

3: $a$ – doubleInput

On entry: the parameter

α

of the gamma distribution.

Constraint:

a > 0.0

4: $b$ – doubleInput

On entry: the parameter

β

of the gamma distribution.

Constraint:

b > 0.0

5: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

On any of the error conditions listed below except $fail . code =$ NE_ALG_NOT_CONV nag_gamma_dist (g01efc) returns $0.0$ .
NE_ALG_NOT_CONV: The algorithm has failed to converge in $〈value〉$ iterations. The probability returned should be a reasonable approximation to the solution.
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_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.

An unexpected error has been triggered by this function. Please contact NAG.
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.
NE_REAL_ARG_LE: On entry, $a = 〈value〉$ and $b = 〈value〉$ .
Constraint: $a > 0.0$ and $b > 0.0$ .
NE_REAL_ARG_LT: On entry, $g = 〈value〉$ .
Constraint: $g \geq 0.0$ .

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. Note also that there is a limit of

18

decimal places on the achievable accuracy, because constants in nag_incomplete_gamma (s14bac) are given to this precision.

8 Parallelism and Performance

nag_gamma_dist (g01efc) is not threaded in any implementation.

9 Further Comments

The time taken by nag_gamma_dist (g01efc) varies slightly with the input arguments g, a and b.

10 Example

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

NAG Library Function Documentnag_gamma_dist (g01efc)

▸▿ 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_gamma_dist (g01efc)