NAG CL Interface
g07abc (ci_​poisson)

Settings help

CL Name Style:

1 Purpose

g07abc computes a confidence interval for the mean parameter of the Poisson distribution.

2 Specification

#include <nag.h>
void  g07abc (Integer n, double xmean, double clevel, double *tl, double *tu, NagError *fail)
The function may be called by the names: g07abc, nag_univar_ci_poisson or nag_poisson_ci.

3 Description

Given a random sample of size n, denoted by x1,x2,,xn, from a Poisson distribution with probability function
p(x)=e-θ θxx! ,  x=0,1,2,  
the point estimate, θ^, for θ is the sample mean, x¯.
Given n and x¯ this function computes a 100(1-α)% confidence interval for the parameter θ, denoted by [θl,θu], where α is in the interval (0,1).
The lower and upper confidence limits are estimated by the solutions to the equations
e-nθlx=T (nθl)xx! =α2, e-nθux=0T(nθu)xx! =α2,  
where T=i=1nxi=nθ^.
The relationship between the Poisson distribution and the χ2-distribution (see page 112 of Hastings and Peacock (1975)) is used to derive the equations
θl= 12n χ2T,α/22, θu= 12n χ2T+2,1-α/22,  
where χν,p2 is the deviate associated with the lower tail probability p of the χ2-distribution with ν degrees of freedom.
In turn the relationship between the χ2-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;
θl= 12n γT,2;α/2, θu= 12n γT+1,2;1-α/2,  
where γα,β;δ is the deviate associated with the lower tail probability, δ, of the gamma distribution with shape parameter α and scale parameter β. These deviates are computed using 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 Integer Input
On entry: n, the sample size.
Constraint: n1.
2: xmean double Input
On entry: the sample mean, x¯.
Constraint: xmean0.0.
3: clevel double Input
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, θl, of the confidence interval.
5: tu double * Output
On exit: the upper limit, θu, of the confidence interval.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge. Both tl and tu are set to zero. This is an unlikely error exit.
On entry, n=value.
Constraint: n>0.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, clevel=value.
Constraint: 0.0<clevel<1.0.
On entry, xmean=value.
Constraint: xmean0.0.

7 Accuracy

For most cases the results should have a relative accuracy of max(0.5e-12,50.0×ε) where ε is the machine precision (see 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 or 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

Background information to multithreading can be found in the Multithreading documentation.
g07abc is not threaded in any implementation.

9 Further Comments


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 parameter θ. g07abc is then called to compute both a 95% and a 99% confidence interval for the parameter θ.

10.1 Program Text

Program Text (g07abce.c)

10.2 Program Data

Program Data (g07abce.d)

10.3 Program Results

Program Results (g07abce.r)