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 9).

4 References

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

5 Arguments

1: $x$ – double Input

On entry:

x

, the value at which the PDF is to be evaluated.

2: $a$ – double Input

On entry:

α

, the shape parameter of the gamma distribution.

Constraint:

a > 0.0

3: $b$ – double Input

On entry:

β

, the scale parameter of the gamma distribution.

Constraints:

$b > 0.0$ ;
$\frac{x}{b} < \frac{1}{nag_real_safe_small_number ()}$ .

4: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

fail . code =

NE_NOERROR, then g01kfc returns

0.0

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_OVERFLOW: Computation abandoned owing to overflow due to extreme parameter values.
NE_REAL: On entry, $a = ⟨ value ⟩$ .
Constraint: $a > 0.0$ .

On entry, $b = ⟨ value ⟩$ .
Constraint: $b > 0.0$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

g01kfc is not threaded in any implementation.

9 Further Comments

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) = \frac{λ^{x}}{x!} e^{- λ} .

(1)

The usual way of computing this quantity would be to take the logarithm and calculate,

\log (x; λ) = x \log λ - \log (x!) - λ .

For large

x

and

λ

x \log λ

and

\log (x!)

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 \times 10^{6}

and

λ = 2 \times 10^{6}

\log (x!) \approx 2.7 \times 10^{7}

and

\log (p (x; λ)) = - 8.17326744645834

. But calculated with the method shown later we have

\log (p (x; λ)) = - 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),

(2)

where

D (x; λ)

, the deviance for the Poisson distribution is given by,

\begin{matrix} D (x; λ) & = & \log (p (x; x)) - \log (p (x; λ)), \\ = & λ D_{0} (\frac{x}{λ}), \end{matrix}

(3)

and

D_{0} (ε) = ε \log ε + 1 - ε .

For

ε

close to

1

D_{0} (ε)

can be evaluated through the series expansion

λ D_{0} (\frac{x}{λ}) = \frac{{(x - λ)}^{2}}{x + λ} + 2 x \sum_{j = 1}^{\infty} \frac{v^{2 j + 1}}{2 j + 1},   where ​ v = \frac{x - λ}{x + λ},

otherwise

D_{0} (ε)

can be evaluated directly. In addition, Loader suggests evaluating

\log (x!)

using the Stirling–De Moivre series,

\log (x!) = \frac{1}{2} \log (2 π x) + x \log (x) - x + δ (x),

(4)

where the error

δ (x)

is given by

δ (x) = \frac{1}{12 x} - \frac{1}{{360 x}^{3}} + \frac{1}{{1260 x}^{5}} + O (x^{−7}) .

Finally

\log (p (x; λ))

can be evaluated by combining equations (1)–(4) to get,

p (x; λ) = \frac{1}{\sqrt{2 π x}} e^{- δ (x) - λ D_{0} (x / λ)} .

10 Example

This example prints the value of the gamma distribution PDF at six different points x with differing a and b.

g01kf: FL CL CPP AD

NAG CL Interfaceg01kfc (pdf_​gamma)

▸▿ 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 CL Interface
g01kfc (pdf_gamma)