NAG Library Function Document

nag_prob_poisson_vector (g01skc)

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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

Not applicable.

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 Documentnag_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

NAG Library Function Document

nag_prob_poisson_vector (g01skc)