void	nag_prob_poisson_vector (Integer ll, const double l[], Integer lk, const Integer k[], double plek[], double pgtk[], double peqk[], Integer ivalid[], NagError *fail)

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}

nag_prob_poisson_vector (g01skc) 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 function 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: $l [ll]$ – const doubleInput

On entry:

λ_{i}

, the parameter of the Poisson distribution with

λ_{i} = l [j]

j = (i - 1) mod ll

, for

i = 1, 2, \dots, \max (ll, lk)

Constraint:

0.0 < l [j - 1] \leq 10^{6}

, for

j = 1, 2, \dots, ll

3: $lk$ – IntegerInput

On entry: the length of the array k.

Constraint:

lk > 0

4: $k [lk]$ – const IntegerInput

On entry:

k_{i}

, the integer which defines the required probabilities with

k_{i} = k [j]

j = (i - 1) mod lk

Constraint:

k [j - 1] \geq 0

, for

j = 1, 2, \dots, lk

5: $plek [\dim]$ – doubleOutput

Note: the dimension, dim, of the array plek must be at least

\max (ll, lk)

On exit:

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

, the lower tail probabilities.

6: $pgtk [\dim]$ – doubleOutput

Note: the dimension, dim, of the array pgtk must be at least

\max (ll, lk)

On exit:

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

, the upper tail probabilities.

7: $peqk [\dim]$ – doubleOutput

Note: the dimension, dim, of the array peqk must be at least

\max (ll, lk)

On exit:

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

, the point probabilities.

8: $ivalid [\dim]$ – IntegerOutput

Note: the dimension, dim, of the array ivalid must be at least

\max (ll, lk)

On exit:

ivalid [i - 1]

indicates any errors with the input arguments, with

$ivalid [i - 1] = 0$

No error.

$ivalid [i - 1] = 1$

On entry,

λ_{i} \leq 0.0

$ivalid [i - 1] = 2$

On entry,

k_{i} < 0

$ivalid [i - 1] = 3$

On entry,

λ_{i} > 10^{6}

9: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ARRAY_SIZE: On entry, $array size = 〈value〉$ .
Constraint: $lk > 0$ .

On entry, $array size = 〈value〉$ .
Constraint: $ll > 0$ .
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NW_IVALID: On entry, at least one value of l or k was invalid.
Check ivalid for more 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

nag_prob_poisson_vector (g01skc) is not threaded in any implementation.

9

Further Comments

The time taken by nag_prob_poisson_vector (g01skc) 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 Function Document

nag_prob_poisson_vector (g01skc)

▸▿ 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