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 <nag.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)

The routine may be called by the names g01skf or nagf_stat_prob_poisson_vector.

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: $ll$ – Integer Input

On entry: the length of the array l.

Constraint:

ll > 0

2: $l (ll)$ – Real (Kind=nag_wp) array Input

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: $lk$ – Integer Input

On entry: the length of the array k.

Constraint:

lk > 0

4: $k (lk)$ – Integer array Input

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) array Output

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) array Output

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) array Output

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 array Output

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: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

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

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.

g01sk: FL CL CPP AD PY MB

NAG FL Interfaceg01skf (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

NAG FL Interface
g01skf (prob_poisson_vector)