NAG Library Routine Document

G01SKF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− 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

G01SKF returns a number of the lower tail, upper tail and point probabilities for the Poisson distribution.

2 Specification

SUBROUTINE G01SKF ( LL, L, LK, K, PLEK, PGTK, PEQK, IVALID, IFAIL)

INTEGER LL, LK, K(LK), IVALID(*), IFAIL

REAL (KIND=nag_wp) L(LL), PLEK(*), PGTK(*), PEQK(*)

3 Description

Let

X = \{X_{i} : i = 1, 2, \dots, m\}

denote a vector of random variables each having a Poisson distribution with parameter

λ_{i}

(> 0)

. Then

Prob \{X_{i} = k_{i}\} = e^{- λ_{i}} \frac{{λ_{i}}^{k_{i}}}{k_{i}!}, k_{i} = 0, 1, 2, \dots

The mean and variance of each distribution are both equal to

λ_{i}

G01SKF computes, for given

λ_{i}

and

k_{i}

the probabilities:

Prob \{X_{i} \leq k_{i}\}

Prob \{X_{i} > k_{i}\}

and

Prob \{X_{i} = k_{i}\}

using the algorithm described in Knüsel (1986).

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

Knüsel L (1986) Computation of the chi-square and Poisson distribution SIAM J. Sci. Statist. Comput. 7 1022–1036

5 Parameters

1: LL – INTEGERInput

On entry: the length of the array L

Constraint:

LL > 0

2: L(LL) – REAL (KIND=nag_wp) arrayInput

On entry:

λ_{i}

, the parameter of the Poisson distribution with

λ_{i} = L (j)

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

, for

i = 1, 2, \dots, \max (LL, LK)

Constraint:

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

, for

j = 1, 2, \dots, LL

3: LK – INTEGERInput

On entry: the length of the array K

Constraint:

LK > 0

4: K(LK) – INTEGER arrayInput

On entry:

k_{i}

, the integer which defines the required probabilities with

k_{i} = K (j)

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

Constraint:

K (j) \geq 0

, for

j = 1, 2, \dots, LK

5: PLEK( $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (LL, LK)

On exit:

Prob \{X_{i} \leq k_{i}\}

, the lower tail probabilities.

6: PGTK( $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (LL, LK)

On exit:

Prob \{X_{i} > k_{i}\}

, the upper tail probabilities.

7: PEQK( $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (LL, LK)

On exit:

Prob \{X_{i} = k_{i}\}

, the point probabilities.

8: IVALID( $*$ ) – INTEGER arrayOutput

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

\max (LL, LK)

On exit:

IVALID (i)

indicates any errors with the input arguments, with

$IVALID (i) = 0$: No error.
$IVALID (i) = 1$: On entry, $λ_{i} \leq 0.0$ .
$IVALID (i) = 2$: On entry, $k_{i} < 0$ .
$IVALID (i) = 3$: On entry, $λ_{i} > 10^{6}$ .

9: 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, if you are not familiar with this parameter, the recommended value is

0

. 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).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, at least one value of L or K was invalid.
Check IVALID for more information.

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

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

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

7 Accuracy

Results are correct to a relative accuracy of at least

10^{- 6}

on machines with a precision of

9

or more decimal digits (provided that the results do not underflow to zero).

8 Further Comments

The time taken by G01SKF to calculate each probability depends on

λ_{i}

and

k_{i}

. For given

λ_{i}

, the time is greatest when

k_{i} \approx λ_{i}

, and is then approximately proportional to

\sqrt{λ_{i}}

9 Example

This example reads a vector of values for

λ

and

k

, and prints the corresponding probabilities.

NAG Library Routine DocumentG01SKF

+− 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

G01SKF