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g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_gamma_pdf (g01kfc)

## 1  Purpose

nag_gamma_pdf (g01kfc) returns the value of the probability density function (PDF) for the gamma distribution with shape argument $\alpha$ and scale argument $\beta$ at a point $x$.

## 2  Specification

 #include #include
 double nag_gamma_pdf (double x, double a, double b, NagError *fail)

## 3  Description

The gamma distribution has PDF
 $fx= 1βαΓα xα-1e-x/β if ​x≥0; α,β>0 fx=0 otherwise.$
If $0.01\le x,\alpha ,\beta \le 100$ 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 8).

## 4  References

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

## 5  Arguments

1:     xdoubleInput
On entry: $x$, the value at which the PDF is to be evaluated.
On entry: $\alpha$, the shape argument of the gamma distribution.
Constraint: ${\mathbf{a}}>0.0$.
3:     bdoubleInput
On entry: $\beta$, the scale argument of the gamma distribution.
Constraints:
• ${\mathbf{b}}>0.0$;
• $\frac{{\mathbf{x}}}{{\mathbf{b}}}<\frac{1}{{\mathbf{nag_real_safe_small_number}}\left(\right)}$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

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_OVERFLOW
Computation abandoned owing to overflow due to extreme parameter values.
NE_REAL
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>0.0$.

## 7  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,
 $px;λ = λx x! e-λ .$ (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
 $logx;λ = x log⁡λ - log x! - λ .$
For large $x$ and $\lambda$, $x\mathrm{log}\lambda$ and $\mathrm{log}\left(x!\right)$ 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 $x=2×{10}^{6}$ and $\lambda =2×{10}^{6}$, $\mathrm{log}\left(x!\right)\approx 2.7×{10}^{7}$ and $\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.17326744645834$. But calculated with the method shown later we have $\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.1732674441334492$. 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,
 $log p x;λ = log p x;x - Dx;λ ,$ (2)
where $D\left(x;\lambda \right)$, the deviance for the Poisson distribution is given by,
 $Dx;λ = log p x;x - log p x;λ , = λ D0 x λ ,$ (3)
and
 $D0 ε = ε log⁡ε + 1 - ε .$
For $\epsilon$ close to $1$, ${D}_{0}\left(\epsilon \right)$ can be evaluated through the series expansion
 $λ D0 x λ = x-λ 2 x+λ + 2x ∑ j=1 ∞ v 2j+1 2j+1 , where ​ v = x-λ x+λ ,$
otherwise ${D}_{0}\left(\epsilon \right)$ can be evaluated directly. In addition, Loader suggests evaluating $\mathrm{log}\left(x!\right)$ using the Stirling–De Moivre series,
 $logx! = 12 log⁡ 2πx + x logx -x + δx ,$ (4)
where the error $\delta \left(x\right)$ is given by
 $δx = 112x - 1 360x3 + 1 1260x5 + O x-7 .$
Finally $\mathrm{log}\left(p\left(x;\lambda \right)\right)$ can be evaluated by combining equations (1)(4) to get,
 $p x;λ = 1 2πx e - δx - λ D0 x/λ .$

## 9  Example

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

### 9.1  Program Text

Program Text (g01kfce.c)

### 9.2  Program Data

Program Data (g01kfce.d)

### 9.3  Program Results

Program Results (g01kfce.r)