NAG C Library Function Document

nag_poisson_ci (g07abc)


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


#include <nag.h>
#include <nagg07.h>
void  nag_poisson_ci (Integer n, double xmean, double clevel, double *tl, double *tu, NagError *fail)


Given a random sample of size n, denoted by x1,x2,,xn, from a Poisson distribution with probability function
px=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 1001-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 nθlxx! =α2, e-nθux=0Tnθuxx! =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 nag_deviates_gamma_dist (g01ffc).


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


1:     n IntegerInput
On entry: n, the sample size.
Constraint: n1.
2:     xmean doubleInput
On entry: the sample mean, x-.
Constraint: xmean0.0.
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, 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 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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.
On entry, clevel=value.
Constraint: 0.0<clevel<1.0.
On entry, xmean=value.
Constraint: xmean0.0.


For most cases the results should have a relative accuracy of max0.5e-12,50.0×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 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.

Parallelism and Performance

nag_poisson_ci (g07abc) is not threaded in any implementation.

Further Comments



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

Program Text

Program Text (g07abce.c)

Program Data

Program Data (g07abce.d)

Program Results

Program Results (g07abce.r)