Integer, Intent (In)	::	ll, lk, k(lk)
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	ivalid(*)
Real (Kind=nag_wp), Intent (In)	::	l(ll)
Real (Kind=nag_wp), Intent (Out)	::	plek(), pgtk(), peqk(*)

C Header Interface

#include nagmk26.h

void	g01skf_ ( const Integer ll, const double l[], const Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], Integer *ifail)

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 arguments 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

Arguments

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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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: $lk > 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

Parallelism and Performance

g01skf is not threaded in any implementation.

9

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}}

10

Example

This example reads a vector of values for

λ

and

k

, and prints the corresponding probabilities.

NAG Library Routine Document

g01skf (prob_poisson_vector)

▸▿ 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

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