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

Syntax

C#
public static double g01ff( double p, double a, double b, double tol, out int ifail )

Visual Basic
Public Shared Function g01ff ( _ p As Double, _ a As Double, _ b As Double, _ tol As Double, _ <OutAttribute> ByRef ifail As Integer _ ) As Double

Visual C++
public: static double g01ff( double p, double a, double b, double tol, [OutAttribute] int% ifail )

F#
static member g01ff : p : float * a : float * b : float * tol : float * ifail : int byref -> float

Parameters

p: Type: System..::..Double
On entry: $p$ , the lower tail probability from the required gamma distribution.

Constraint: $0.0 \leq p < 1.0$ .

a: Type: System..::..Double
On entry: $α$ , the shape parameter of the gamma distribution.

Constraint: $0.0 < a \leq 10^{6}$ .

b: Type: System..::..Double
On entry: $β$ , the scale parameter of the gamma distribution.

Constraint: $b > 0.0$ .

tol: Type: System..::..Double
On entry: 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 g01ff is entered with tol less than $50 \times δ$ or greater or equal to $1.0$ , then $50 \times δ$ is used instead.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

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

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

Error Indicators and Warnings

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

Errors or warnings detected by the method:

If on exit

ifail = 1

2

3

5

, then g01ff returns

0.0

$ifail = 1$

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.

$ifail = 4$: 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.

$ifail = -9000$: An error occured, see message report.

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.

Parallelism and Performance

None.

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.

Example program (C#): g01ffe.cs

Example program data: g01ffe.d

Example program results: g01ffe.r