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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
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

subsamples 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)