NAG Library Function Document

nag_gamma_pdf_vector (g01kkc)

void	nag_gamma_pdf_vector (Nag_Boolean ilog, Integer lx, const double x[], Integer la, const double a[], Integer lb, const double b[], double pdf[], Integer ivalid[], NagError *fail)

3 Description

The gamma distribution with shape parameter

α_{i}

and scale parameter

β_{i}

has PDF

\begin{matrix} f (x_{i}, α_{i}, β_{i}) = \frac{1}{{β_{i}}^{α_{i}} Γ (α_{i})} {x_{i}}^{α_{i} - 1} e^{- x_{i} / β_{i}} & if ​ x_{i} \geq 0; α_{i}, β_{i} > 0 \\ f (x_{i}, α_{i}, β_{i}) = 0 & otherwise. \end{matrix}

0.01 \leq x_{i}, α_{i}, β_{i} \leq 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 9).

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

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

5 Arguments

1: ilog – Nag_BooleanInput

On entry: the value of ilog determines whether the logarithmic value is returned in pdf.

$ilog = Nag_FALSE$: $f (x_{i}, α_{i}, β_{i})$ , the probability density function is returned.
$ilog = Nag_TRUE$: $\log (f (x_{i}, α_{i}, β_{i}))$ , the logarithm of the probability density function is returned.

2: lx – IntegerInput

On entry: the length of the array x.

Constraint:

lx > 0

3: x[lx] – const doubleInput

On entry:

x_{i}

, the values at which the PDF is to be evaluated with

x_{i} = x [j]

j = (i - 1) mod lx

, for

i = 1, 2, \dots, \max (lx, la, lb)

4: la – IntegerInput

On entry: the length of the array a.

Constraint:

la > 0

5: a[la] – const doubleInput

On entry:

α_{i}

, the shape parameter with

α_{i} = a [j]

j = (i - 1) mod la

Constraint:

a [j - 1] > 0.0

, for

j = 1, 2, \dots, la

6: lb – IntegerInput

On entry: the length of the array b.

Constraint:

lb > 0

7: b[lb] – const doubleInput

On entry:

β_{i}

, the scale parameter with

β_{i} = b [j]

j = (i - 1) mod lb

Constraint:

b [j - 1] > 0.0

, for

j = 1, 2, \dots, lb

8: pdf[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array pdf must be at least

\max (lx, la, lb)

On exit:

f (x_{i}, α_{i}, β_{i})

\log (f (x_{i}, α_{i}, β_{i}))

9: ivalid[ $\dim$ ] – IntegerOutput

Note: the dimension, dim, of the array ivalid must be at least

\max (lx, la, lb)

On exit:

ivalid [i - 1]

indicates any errors with the input arguments, with

$ivalid [i - 1] = 0$: No error.
$ivalid [i - 1] = 1$: $α_{i} \leq 0.0$ .
$ivalid [i - 1] = 2$: $β_{i} \leq 0.0$ .
$ivalid [i - 1] = 3$: $\frac{x_{i}}{β_{i}}$ overflows, the value returned should be a reasonable approximation.

10: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ARRAY_SIZE: On entry, $array size = ⟨value⟩$ .
Constraint: $la > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $lb > 0$ .
On entry, $array size = ⟨value⟩$ .
Constraint: $lx > 0$ .
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.
NW_IVALID: On entry, at least one value of x, a or b was invalid.
Check ivalid for more information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Not applicable.

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 (p (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_{i}

with differing

α_{i}

and

β_{i}

NAG Library Function Documentnag_gamma_pdf_vector (g01kkc)

+− 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_pdf_vector (g01kkc)