g01ff: FL CL CPP AD PY MB

NAG CL Interface
g01ffc (inv_cdf_gamma)

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NAG Library Manual, Mark 29.2

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g01ff: FL CL CPP AD PY MB

▸▿ 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

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1 Purpose

g01ffc returns the deviate associated with the given lower tail probability of the gamma distribution.

2 Specification

#include <nag.h>

double

g01ffc (double p, double a, double b, double tol, NagError *fail)

The function may be called by the names: g01ffc, nag_stat_inv_cdf_gamma or nag_deviates_gamma_dist.

3 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.

4 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

5 Arguments

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

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

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

Constraint: $b > 0.0$ .
4: $tol$ – double Input: 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 g01ffc is entered with tol less than $50 \times δ$ or greater or equal to $1.0$ , then $50 \times δ$ is used instead.
5: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If on exit

fail . code =

NE_GAM_NOT_CONV, NE_PROBAB_CLOSE_TO_TAIL, NE_REAL_ARG_GE, NE_REAL_ARG_GT, NE_REAL_ARG_LE or NE_REAL_ARG_LT, then g01ffc returns

0.0

On any of the error conditions listed below, except $fail . code =$ NE_ALG_NOT_CONV, g01ffc returns $0.0$ .
NE_ALG_NOT_CONV: The algorithm has failed to converge in $100$ iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
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_GAM_NOT_CONV: The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
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_PROBAB_CLOSE_TO_TAIL: The probability is too close to $0.0$ for the given a to enable the result to be calculated.
NE_REAL_ARG_GE: On entry, $p = ⟨ value ⟩$ .
Constraint: $p < 1.0$ .
NE_REAL_ARG_GT: On entry, $a = ⟨ value ⟩$ .
Constraint: $a \leq 10^{6}$ .
NE_REAL_ARG_LE: On entry, $a = ⟨ value ⟩$ .
Constraint: $a > 0.0$ .

On entry, $b = ⟨ value ⟩$ .
Constraint: $b > 0.0$ .
NE_REAL_ARG_LT: On entry, $p = ⟨ value ⟩$ .
Constraint: $p \geq 0.0$ .

7 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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g01ffc is not threaded in any implementation.

9 Further Comments

None.

10 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.

g01ff: FL CL CPP AD PY MB

NAG CL Interfaceg01ffc (inv_​cdf_​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
g01ffc (inv_cdf_gamma)