NAG Library Routine Document

G01TFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01TFF returns a number of deviates associated with given probabilities of the gamma distribution.

2 Specification

SUBROUTINE G01TFF (

LTAIL, TAIL, LP, P, LA, A, LB, B, TOL, G, IVALID, IFAIL)

INTEGER	LTAIL, LP, LA, LB, IVALID(*), IFAIL
REAL (KIND=nag_wp)	P(LP), A(LA), B(LB), TOL, G(*)
CHARACTER(1)	TAIL(LTAIL)

3 Description

The deviate,

g_{p_{i}}

, associated with the lower tail probability,

p_{i}

, of the gamma distribution with shape parameter

α_{i}

and scale parameter

β_{i}

, is defined as the solution to

P (G_{i} \leq g_{p_{i}} : α_{i}, β_{i}) = p_{i} = \frac{1}{{β_{i}}^{α_{i}} Γ (α_{i})} \int_{0}^{g_{p_{i}}} {e_{i}}^{- G_{i} / β_{i}} {G_{i}}^{α_{i} - 1} d G_{i}, 0 \leq g_{p_{i}} < \infty; ​ α_{i}, β_{i} > 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_{i} = 2 \frac{g_{p_{i}}}{β_{i}}

. The required

y_{i}

is found from the Taylor series expansion

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

where

y_{0}

is a starting approximation

$C_{1} (u_{i}) = 1$ ,
$C_{r + 1} (u_{i}) = (r Ψ + \frac{d}{d u_{i}}) C_{r} (u_{i})$ ,
$Ψ_{i} = \frac{1}{2} - \frac{α_{i} - 1}{u_{i}}$ ,
$E_{i} = p_{i} - \int_{0}^{y_{0}} ϕ_{i} (u_{i}) d u_{i}$ ,
$ϕ_{i} (u_{i}) = \frac{1}{2^{α_{i}} Γ (α_{i})} {e_{i}}^{- u_{i} / 2} {u_{i}}^{α_{i} - 1}$ .

For most values of

p_{i}

and

α_{i}

the starting value

y_{01} = 2 α_{i} {(z_{i} \sqrt{\frac{1}{9 α_{i}}} + 1 - \frac{1}{9 α_{i}})}^{3}

is used, where

z_{i}

is the deviate associated with a lower tail probability of

p_{i}

for the standard Normal distribution.

For

p_{i}

close to zero,

y_{02} = {(p_{i} α_{i} 2^{α_{i}} Γ (α_{i}))}^{1 / α_{i}}

is used.

For large

p_{i}

values, when

y_{01} > 4.4 α_{i} + 6.0

y_{03} = - 2 [\ln (1 - p_{i}) - (α_{i} - 1) \ln (\frac{1}{2} y_{01}) + \ln (Γ (α_{i}))]

is found to be a better starting value than

y_{01}

For small

α_{i}

(α_{i} \leq 0.16)

p_{i}

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.

The input arrays to this routine are designed to allow maximum flexibility in the supply of vector parameters 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

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 Parameters

1: LTAIL – INTEGERInput

On entry: the length of the array TAIL.

Constraint:

LTAIL > 0

2: TAIL(LTAIL) – CHARACTER(1) arrayInput

On entry: indicates which tail the supplied probabilities represent. For

j = ((i - 1) mod LTAIL) + 1

, for

i = 1, 2, \dots, \max (LTAIL, LP, LA, LB)

$TAIL (j) ='L'$: The lower tail probability, i.e., $p_{i} = P (G_{i} \leq g_{p_{i}} : α_{i}, β_{i})$ .
$TAIL (j) ='U'$: The upper tail probability, i.e., $p_{i} = P (G_{i} \geq g_{p_{i}} : α_{i}, β_{i})$ .

Constraint:

TAIL (j) ='L'

'U'

, for

j = 1, 2, \dots, LTAIL

3: LP – INTEGERInput

On entry: the length of the array P.

Constraint:

LP > 0

4: P(LP) – REAL (KIND=nag_wp) arrayInput

On entry:

p_{i}

, the probability of the required gamma distribution as defined by TAIL with

p_{i} = P (j)

j = ((i - 1) mod LP) + 1

Constraints:

if $TAIL (k) ='L'$ , $0.0 \leq P (j) < 1.0$ ;
otherwise $0.0 < P (j) \leq 1.0$ .

Where

k = (i - 1) mod LTAIL + 1

and

j = (i - 1) mod LP + 1

5: LA – INTEGERInput

On entry: the length of the array A.

Constraint:

LA > 0

6: A(LA) – REAL (KIND=nag_wp) arrayInput

On entry:

α_{i}

, the first parameter of the required gamma distribution with

α_{i} = A (j)

j = ((i - 1) mod LA) + 1

Constraint:

0.0 < A (j) \leq 10^{6}

, for

j = 1, 2, \dots, LA

7: LB – INTEGERInput

On entry: the length of the array B.

Constraint:

LB > 0

8: B(LB) – REAL (KIND=nag_wp) arrayInput

On entry:

β_{i}

, the second parameter of the required gamma distribution with

β_{i} = B (j)

j = ((i - 1) mod LB) + 1

Constraint:

B (j) > 0.0

, for

j = 1, 2, \dots, LB

9: TOL – REAL (KIND=nag_wp)Input

On entry: the relative accuracy required by you in the results. If G01TFF is entered with TOL greater than or equal to

1.0

or less than

10 \times machine precision

(see X02AJF), then the value of

10 \times machine precision

is used instead.

10: G( $*$ ) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array G must be at least

\max (LTAIL, LP, LA, LB)

On exit:

g_{p_{i}}

, the deviates for the gamma distribution.

11: IVALID( $*$ ) – INTEGER arrayOutput

Note: the dimension of the array IVALID must be at least

\max (LTAIL, LP, LA, LB)

On exit:

IVALID (i)

indicates any errors with the input arguments, with

$IVALID (i) = 0$

No error.

$IVALID (i) = 1$

On entry,

invalid value supplied in TAIL when calculating

g_{p_{i}}

$IVALID (i) = 2$

On entry,

invalid value for

p_{i}

$IVALID (i) = 3$

On entry,	$α_{i} \leq 0.0$ ,
or	$α_{i} > 10^{6}$ ,
or	$β_{i} \leq 0.0$ .

$IVALID (i) = 4$

p_{i}

is too close to

0.0

1.0

to enable the result to be calculated.

$IVALID (i) = 5$

The solution has failed to converge. The result may be a reasonable approximation.

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: G01TFF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, at least one value of TAIL, P, A, or B was invalid.
Check IVALID for more information.

$IFAIL = 2$: On entry, $array size = ⟨value⟩$ .
Constraint: $LTAIL > 0$ .

$IFAIL = 3$: On entry, $array size = ⟨value⟩$ .
Constraint: $LP > 0$ .

$IFAIL = 4$: On entry, $array size = ⟨value⟩$ .
Constraint: $LA > 0$ .

$IFAIL = 5$: On entry, $array size = ⟨value⟩$ .
Constraint: $LB > 0$ .

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

7 Accuracy

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

α_{i}

or very small values of

p_{i}

there may be some loss of accuracy.

8 Further Comments

None.

9 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.

NAG Library Routine DocumentG01TFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G01TFF