NAG Library Function Document

nag_poisson_ci (g07abc)

+− 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

nag_poisson_ci (g07abc) computes a confidence interval for the mean argument of the Poisson distribution.

2 Specification

#include <nag.h>

#include <nagg07.h>

void	nag_poisson_ci (Integer n, double xmean, double clevel, double tl, double tu, NagError *fail)

3 Description

Given a random sample of size

n

, denoted by

x_{1}, x_{2}, \dots, x_{n}

, from a Poisson distribution with probability function

p (x) = e^{- θ} \frac{θ^{x}}{x!}, x = 0, 1, 2, \dots

the point estimate,

\hat{θ}

, for

θ

is the sample mean,

\bar{x}

Given

n

and

\bar{x}

this function computes a

100 (1 - α) %

confidence interval for the argument

θ

, denoted by [

θ_{l}, θ_{u}

], where

α

is in the interval

(0, 1)

The lower and upper confidence limits are estimated by the solutions to the equations

\begin{matrix} e^{- n θ_{l}} \sum_{x = T}^{\infty} \frac{{(n θ_{l})}^{x}}{x!} = \frac{α}{2}, \\ e^{- n θ_{u}} \sum_{x = 0}^{T} \frac{{(n θ_{u})}^{x}}{x!} = \frac{α}{2}, \end{matrix}

where

T = \sum_{i = 1}^{n} x_{i} = n \hat{θ}

The relationship between the Poisson distribution and the

χ^{2}

-distribution (see page 112 of Hastings and Peacock (1975)) is used to derive the equations

\begin{array}{l} θ_{l} = \frac{1}{2 n} χ_{2 T, α / 2}^{2}, \\ θ_{u} = \frac{1}{2 n} χ_{2 T + 2, 1 - α / 2}^{2}, \end{array}

where

χ_{ν, p}^{2}

is the deviate associated with the lower tail probability

p

of the

χ^{2}

-distribution with

ν

degrees of freedom.

In turn the relationship between the

χ^{2}

-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;

\begin{array}{l} θ_{l} = \frac{1}{2 n} γ_{T, 2; α / 2}, \\ θ_{u} = \frac{1}{2 n} γ_{T + 1, 2; 1 - α / 2}, \end{array}

where

γ_{α, β; δ}

is the deviate associated with the lower tail probability,

δ

, of the gamma distribution with shape argument

α

and scale argument

β

. These deviates are computed using nag_deviates_gamma_dist (g01ffc).

4 References

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press

5 Arguments

1: n – IntegerInput: On entry: $n$ , the sample size.
Constraint: $n \geq 1$ .
2: xmean – doubleInput: On entry: the sample mean, $\bar{x}$ .
Constraint: $xmean \geq 0.0$ .
3: clevel – doubleInput: On entry: the confidence level, $(1 - α)$ , for two-sided interval estimate. For example $clevel = 0.95$ gives a $95 %$ confidence interval.
Constraint: $0.0 < clevel < 1.0$ .
4: tl – double *Output: On exit: the lower limit, $θ_{l}$ , of the confidence interval.
5: tu – double *Output: On exit: the upper limit, $θ_{u}$ , of the confidence interval.
6: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_CONVERGENCE: When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n > 0$ .
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.
NE_REAL: On entry, $clevel \leq 0.0$ or $clevel \geq 1.0$ : $clevel = ⟨value⟩$ .
On entry, $xmean = ⟨value⟩$ .
Constraint: $xmean \geq 0.0$ .

7 Accuracy

For most cases the results should have a relative accuracy of

\max (0.5e - 12, 50.0 \times ε)

where

ε

is the machine precision (see nag_machine_precision (X02AJC)). Thus on machines with sufficiently high precision the results should be accurate to

12

significant digits. Some accuracy may be lost when

α / 2

1 - α / 2

is very close to

0.0

, which will occur if clevel is very close to

1.0

. This should not affect the usual confidence intervals used.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

The following example reads in data showing the number of noxious weed seeds and the frequency with which that number occurred in

98

sub-samples of meadow grass. The data is taken from page 224 of Snedecor and Cochran (1967). The sample mean is computed as the point estimate of the Poisson argument

θ

. nag_poisson_ci (g07abc) is then called to compute both a 95% and a 99% confidence interval for the argument

θ

NAG Library Function Documentnag_poisson_ci (g07abc)

+− 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_poisson_ci (g07abc)