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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_gamma (g01ff)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_stat_inv_cdf_gamma (g01ff) returns the deviate associated with the given lower tail probability of the gamma distribution.

Syntax

[result, ifail] = g01ff(p, a, b, 'tol', tol)

[result, ifail] = nag_stat_inv_cdf_gamma(p, a, b, 'tol', tol)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 23:

tol was made optional (default 0)

Description

The deviate,

g_{p}

, associated with the lower tail probability,

p

, of the gamma distribution with shape parameter

α

and scale parameter

β

, is defined as the solution to

P (G \leq g_{p} : α, β) = p = \frac{1}{β^{α} Γ (α)} \int_{0}^{g_{p}} e^{- G / β} G^{α - 1} d G, 0 \leq g_{p} < \infty; α, β > 0 .

The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the

χ^{2}

-distribution.

Let

y = 2 \frac{g_{p}}{β}

. The required

y

is found from the Taylor series expansion

y = y_{0} + \sum_{r} \frac{C_{r} (y_{0})}{r!} {(\frac{E}{ϕ (y_{0})})}^{r},

where

y_{0}

is a starting approximation

$C_{1} (u) = 1$ ,
$C_{r + 1} (u) = (r Ψ + \frac{d}{d u}) C_{r} (u)$ ,
$Ψ = \frac{1}{2} - \frac{α - 1}{u}$ ,
$E = p - \int_{0}^{y_{0}} ϕ (u) d u$ ,
$ϕ (u) = \frac{1}{2^{α} Γ (α)} e^{- u / 2} u^{α - 1}$ .

For most values of

p

and

α

the starting value

y_{01} = 2 α {(z \sqrt{\frac{1}{9 α}} + 1 - \frac{1}{9 α})}^{3}

is used, where

z

is the deviate associated with a lower tail probability of

p

for the standard Normal distribution.

For

p

close to zero,

y_{02} = {(p α 2^{α} Γ (α))}^{1 / α}

is used.

For large

p

values, when

y_{01} > 4.4 α + 6.0

y_{03} = - 2 [\ln (1 - p) - (α - 1) \ln (\frac{1}{2} y_{01}) + \ln (Γ (α))]

is found to be a better starting value than

y_{01}

For small

α

(α \leq 0.16)

p

is expressed in terms of an approximation to the exponential integral and

y_{04}

is found by Newton–Raphson iterations.

Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.

References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the

χ^{2}

distribution Appl. Statist. 24 385–388

Parameters

Compulsory Input Parameters

1: $p$ – double scalar: $p$ , the lower tail probability from the required gamma distribution.

Constraint: $0.0 \leq p < 1.0$ .
2: $a$ – double scalar: $α$ , the shape parameter of the gamma distribution.

Constraint: $0.0 < a \leq 10^{6}$ .
3: $b$ – double scalar: $β$ , the scale parameter of the gamma distribution.

Constraint: $b > 0.0$ .

Optional Input Parameters

1: $tol$ – double scalar: Default: $0.0$
The relative accuracy required by you in the results. The smallest recommended value is $50 \times δ$ , where $δ = \max (10^{- 18}, machine precision)$ . If nag_stat_inv_cdf_gamma (g01ff) is entered with tol less than $50 \times δ$ or greater or equal to $1.0$ , then $50 \times δ$ is used instead.

Output Parameters

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

Error Indicators and Warnings

Note: nag_stat_inv_cdf_gamma (g01ff) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

If on exit

ifail = 1

2

3

5

, then nag_stat_inv_cdf_gamma (g01ff) returns

0.0

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$p < 0.0$ ,
or	$p \geq 1.0$ ,

$ifail = 2$

On entry,	$a \leq 0.0$ ,
or	$a > 10^{6}$ ,
or	$b \leq 0.0$

$ifail = 3$: p is too close to $0.0$ or $1.0$ to enable the result to be calculated.

W $ifail = 4$: The solution has failed to converge in $100$ iterations. A larger value of tol should be tried. The result may be a reasonable approximation.

$ifail = 5$: The series to calculate the gamma function has failed to converge. This is an unlikely error exit.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of

α

or very small values of

p

there may be some loss of accuracy.

Further Comments

None.

Example

This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.

Open in the MATLAB editor: g01ff_example

function g01ff_example


fprintf('g01ff example results\n\n');

p = [  0.01    0.428   0.869 ];
a = [  1       7.5    45     ];
b = [ 20       0.1    10     ];
x  = p;

fprintf('     p       a       b       x\n');
for j = 1:numel(p)
   [x(j), ifail] = g01ff( ...
			    p(j), a(j), b(j));
end

fprintf('%8.3f%8.3f%8.3f%8.3f\n', [p; a; b; x]);

g01ff example results

     p       a       b       x
   0.010   1.000  20.000   0.201
   0.428   7.500   0.100   0.670
   0.869  45.000  10.000 525.839

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Chapter Contents

Chapter Introduction

NAG Toolbox