g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_gamma_vector (g01sfc)

## 1  Purpose

nag_prob_gamma_vector (g01sfc) returns a number of lower or upper tail probabilities for the gamma distribution.

## 2  Specification

 #include #include
 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 ${\alpha }_{i}$ and ${\beta }_{i}$, $P\left({G}_{i}\le {g}_{i}\right)$, is defined by:
 $P Gi ≤ gi :αi,βi = 1 βi αi Γ αi ∫ 0 gi Gi αi-1 e -Gi/βi dGi , αi>0.0 , ​ βi>0.0 .$
The mean of the distribution is ${\alpha }_{i}{\beta }_{i}$ and its variance is ${\alpha }_{i}{{\beta }_{i}}^{2}$. The transformation ${Z}_{i}=\frac{{G}_{i}}{{\beta }_{i}}$ is applied to yield the following incomplete gamma function in normalized form,
 $P Gi ≤ gi :αi,βi = P Zi ≤ gi / βi :αi,1.0 = 1 Γ αi ∫ 0 gi / βi Zi αi-1 e -Zi dZi .$
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:    $\mathbf{ltail}$IntegerInput
On entry: the length of the array tail.
Constraint: ${\mathbf{ltail}}>0$.
2:    $\mathbf{tail}\left[{\mathbf{ltail}}\right]$const Nag_TailProbabilityInput
On entry: indicates whether a lower or upper tail probability is required. For , for $\mathit{i}=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{ltail}},{\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}}\right)$:
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_LowerTail}$
The lower tail probability is returned, i.e., ${p}_{i}=P\left({G}_{i}\le {g}_{i}:{\alpha }_{i},{\beta }_{i}\right)$.
${\mathbf{tail}}\left[j\right]=\mathrm{Nag_UpperTail}$
The upper tail probability is returned, i.e., ${p}_{i}=P\left({G}_{i}\ge {g}_{i}:{\alpha }_{i},{\beta }_{i}\right)$.
Constraint: ${\mathbf{tail}}\left[\mathit{j}-1\right]=\mathrm{Nag_LowerTail}$ or $\mathrm{Nag_UpperTail}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ltail}}$.
3:    $\mathbf{lg}$IntegerInput
On entry: the length of the array g.
Constraint: ${\mathbf{lg}}>0$.
4:    $\mathbf{g}\left[{\mathbf{lg}}\right]$const doubleInput
On entry: ${g}_{i}$, the value of the gamma variate with ${g}_{i}={\mathbf{g}}\left[j\right]$, .
Constraint: ${\mathbf{g}}\left[\mathit{j}-1\right]\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lg}}$.
5:    $\mathbf{la}$IntegerInput
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
6:    $\mathbf{a}\left[{\mathbf{la}}\right]$const doubleInput
On entry: the parameter ${\alpha }_{i}$ of the gamma distribution with ${\alpha }_{i}={\mathbf{a}}\left[j\right]$, .
Constraint: ${\mathbf{a}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
7:    $\mathbf{lb}$IntegerInput
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
8:    $\mathbf{b}\left[{\mathbf{lb}}\right]$const doubleInput
On entry: the parameter ${\beta }_{i}$ of the gamma distribution with ${\beta }_{i}={\mathbf{b}}\left[j\right]$, .
Constraint: ${\mathbf{b}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
9:    $\mathbf{p}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array p must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}},{\mathbf{ltail}}\right)$.
On exit: ${p}_{i}$, the probabilities of the beta distribution.
10:  $\mathbf{ivalid}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lg}},{\mathbf{la}},{\mathbf{lb}},{\mathbf{ltail}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
 On entry, invalid value supplied in tail when calculating ${p}_{i}$.
${\mathbf{ivalid}}\left[i-1\right]=2$
 On entry, ${g}_{i}<0.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
 On entry, ${\alpha }_{i}\le 0.0$, or ${\beta }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=4$
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:  $\mathbf{fail}$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.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lg}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ltail}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ 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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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.

## 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 $1$ or $2$ decimal places.

## 8  Parallelism and Performance

Not applicable.

The time taken by nag_prob_gamma_vector (g01sfc) to calculate each probability varies slightly with the input arguments ${g}_{i}$, ${\alpha }_{i}$ and ${\beta }_{i}$.

## 10  Example

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

### 10.1  Program Text

Program Text (g01sfce.c)

### 10.2  Program Data

Program Data (g01sfce.d)

### 10.3  Program Results

Program Results (g01sfce.r)