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 3.6 in the Essential Introduction).

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_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.
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

Not applicable.

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)