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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_pdf_gamma (g01kf)

## Purpose

nag_stat_pdf_gamma (g01kf) returns the value of the probability density function (PDF) for the gamma distribution with shape parameter α$\alpha$ and scale parameter β$\beta$ at a point x$x$.

## Syntax

[result, ifail] = g01kf(x, a, b)
[result, ifail] = nag_stat_pdf_gamma(x, a, b)

## Description

The gamma distribution has PDF
 f(x) = 1/(βαΓ(α))xα − 1e − x / β if ​x ≥ 0;  α,β > 0 f(x) = 0 otherwise.
$f(x)= 1βαΓ(α) xα-1e-x/β if ​x≥0; α,β>0 f(x)=0 otherwise.$
If 0.01x,α,β100$0.01\le x,\alpha ,\beta \le 100$ then an algorithm based directly on the gamma distribution's PDF is used. For values outside this range, the function is calculated via the Poisson distribution's PDF as described in Loader (2000) (see Section [Further Comments]).

## References

Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)

## Parameters

### Compulsory Input Parameters

1:     x – double scalar
x$x$, the value at which the PDF is to be evaluated.
2:     a – double scalar
α$\alpha$, the shape parameter of the gamma distribution.
Constraint: a > 0.0${\mathbf{a}}>0.0$.
3:     b – double scalar
β$\beta$, the scale parameter of the gamma distribution.
Constraints:
• b > 0.0${\mathbf{b}}>0.0$;
• (x)/(b) < 1/(x02am())$\frac{{\mathbf{x}}}{{\mathbf{b}}}<\frac{1}{\mathbf{x02am}\left(\right)}$.

None.

None.

### Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
If ${\mathbf{ifail}}\ne {\mathbf{0}}$, then nag_stat_pdf_gamma (g01kf) returns 0.0$0.0$.
ifail = 1${\mathbf{ifail}}=1$
Constraint: a > 0.0${\mathbf{a}}>0.0$.
ifail = 2${\mathbf{ifail}}=2$
Constraint: b > 0.0${\mathbf{b}}>0.0$.
ifail = 3${\mathbf{ifail}}=3$
Computation abandoned owing to overflow due to extreme parameter values.

## Accuracy

Not applicable.

Due to the lack of a stable link to Loader (2000) paper, we give a brief overview of the method, as applied to the Poisson distribution. The Poisson distribution has a continuous mass function given by,
 p(x ; λ) = (λx)/(x ! ) e − λ . $p(x;λ) = λx x! e-λ .$ (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
 log(x ; λ) = x logλ − log(x ! ) − λ . $log(x;λ) = x log⁡λ - log( x! ) - λ .$
For large x$x$ and λ$\lambda$, xlogλ$x\mathrm{log}\lambda$ and log(x ! )$\mathrm{log}\left(x!\right)$ are very large, of the same order of magnitude and when calculated have rounding errors. The subtraction of these two terms can therefore result in a number, many orders of magnitude smaller and hence we lose accuracy due to subtraction errors. For example for x = 2 × 106$x=2×{10}^{6}$ and λ = 2 × 106$\lambda =2×{10}^{6}$, log(x ! )2.7 × 107$\mathrm{log}\left(x!\right)\approx 2.7×{10}^{7}$ and log(p(x ; λ)) = 8.17326744645834$\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.17326744645834$. But calculated with the method shown later we have log(p(x ; λ)) = 8.1732674441334492$\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.1732674441334492$. The difference between these two results suggests a loss of about 7 significant figures of precision.
Loader introduces an alternative way of expressing (1) based on the saddle point expansion,
 log(p(x ; λ)) = log(p(x ; x)) − D(x ; λ) , $log( p (x;λ) ) = log( p (x;x) ) - D(x;λ) ,$ (2)
where D(x ; λ)$D\left(x;\lambda \right)$, the deviance for the Poisson distribution is given by,
 D(x ; λ) = log(p(x ; x)) − log(p(x ; λ)) , = λ D0 (x/λ) ,
$D(x;λ) = log( p (x;x) ) - log( p (x;λ) ) , = λ D0 ( x λ ) ,$
(3)
and
 D0 (ε) = ε logε + 1 − ε . $D0 (ε) = ε log⁡ε + 1 - ε .$
For ε$\epsilon$ close to 1$1$, D0(ε)${D}_{0}\left(\epsilon \right)$ can be evaluated through the series expansion
 ∞ λD0(x/λ) = ((x − λ)2)/(x + λ) + 2x ∑ (v2j + 1)/(2j + 1) ,  where ​v = (x − λ)/(x + λ), j = 1
$λ D0 ( x λ ) = (x-λ) 2 x+λ + 2x ∑ j=1 ∞ v 2j+1 2j+1 , where ​ v = x-λ x+λ ,$
otherwise D0(ε)${D}_{0}\left(\epsilon \right)$ can be evaluated directly. In addition, Loader suggests evaluating log(x ! )$\mathrm{log}\left(x!\right)$ using the Stirling–De Moivre series,
 log(x ! ) = (1/2) log (2πx) + x log(x) − x + δ(x) , $log(x!) = 12 log⁡ (2πx) + x log(x) -x + δ(x) ,$ (4)
where the error δ(x)$\delta \left(x\right)$ is given by
 δ(x) = 1/(12x) − 1/(360x3) + 1/(1260x5) + O (x − 7) . $δ(x) = 112x - 1 360x3 + 1 1260x5 + O ( x-7 ) .$
Finally log(p(x ; λ))$\mathrm{log}\left(p\left(x;\lambda \right)\right)$ can be evaluated by combining equations (1)(4) to get,
 p (x ; λ) = 1/(sqrt(2πx)) e − δ(x) − λ D0 (x / λ) . $p (x;λ) = 1 2πx e - δ(x) - λ D0 ( x/λ ) .$

## Example

```function nag_stat_pdf_gamma_example
x      = [0.1, 3, 6, 4, 9, 16];
a      = [3, 10, 5, 10, 9, 3.5];
b      = [2, 11, 1, 0.1, 0.5, 2.5];
result = zeros(6, 1);
fprintf('\n      X            A            B          Result\n');

for i=1:6
[result(i), ifail] = nag_stat_pdf_gamma(x(i), a(i), b(i));
fprintf('%12.4e %12.4e %12.4e %12.4e\n', x(i), a(i), b(i), result(i));
end
```
```

X            A            B          Result
1.0000e-01   3.0000e+00   2.0000e+00   5.9452e-04
3.0000e+00   1.0000e+01   1.1000e+01   1.5921e-12
6.0000e+00   5.0000e+00   1.0000e+00   1.3385e-01
4.0000e+00   1.0000e+01   1.0000e-01   3.0690e-08
9.0000e+00   9.0000e+00   5.0000e-01   8.3251e-03
1.6000e+01   3.5000e+00   2.5000e+00   2.0723e-02

```
```function g01kf_example
x      = [0.1, 3, 6, 4, 9, 16];
a      = [3, 10, 5, 10, 9, 3.5];
b      = [2, 11, 1, 0.1, 0.5, 2.5];
result = zeros(6, 1);
fprintf('\n      X            A            B          Result\n');

for i=1:6
[result(i), ifail] = g01kf(x(i), a(i), b(i));
fprintf('%12.4e %12.4e %12.4e %12.4e\n', x(i), a(i), b(i), result(i));
end
```
```

X            A            B          Result
1.0000e-01   3.0000e+00   2.0000e+00   5.9452e-04
3.0000e+00   1.0000e+01   1.1000e+01   1.5921e-12
6.0000e+00   5.0000e+00   1.0000e+00   1.3385e-01
4.0000e+00   1.0000e+01   1.0000e-01   3.0690e-08
9.0000e+00   9.0000e+00   5.0000e-01   8.3251e-03
1.6000e+01   3.5000e+00   2.5000e+00   2.0723e-02

```