NAG Library Function Document
nag_poisson_ci (g07abc) computes a confidence interval for the mean argument of the Poisson distribution.
||nag_poisson_ci (Integer n,
Given a random sample of size
, denoted by
, from a Poisson distribution with probability function
the point estimate,
is the sample mean,
Given and this function computes a confidence interval for the argument , denoted by , where is in the interval .
The lower and upper confidence limits are estimated by the solutions to the equations
The relationship between the Poisson distribution and the
-distribution (see page 112 of Hastings and Peacock (1975)
) is used to derive the equations
is the deviate associated with the lower tail probability
degrees of freedom.
In turn the relationship between the
-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)
) yields the following equivalent equations;
is the deviate associated with the lower tail probability,
, of the gamma distribution with shape argument
and scale argument
. 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
n – IntegerInput
On entry: , the sample size.
xmean – doubleInput
On entry: the sample mean, .
clevel – doubleInput
On entry: the confidence level, , for two-sided interval estimate. For example gives a confidence interval.
tl – double *Output
On exit: the lower limit, , of the confidence interval.
tu – double *Output
On exit: the upper limit, , of the confidence interval.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, argument had an illegal value.
When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge.
On entry, .
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
On entry, or : .
On entry, .
For most cases the results should have a relative accuracy of
is the machine precision
(see nag_machine_precision (X02AJC)
). Thus on machines with sufficiently high precision the results should be accurate to
significant digits. Some accuracy may be lost when
is very close to
, which will occur if clevel
is very close to
. This should not affect the usual confidence intervals used.
The following example reads in data showing the number of noxious weed seeds and the frequency with which that number occurred in
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
. nag_poisson_ci (g07abc) is then called to compute both a 95% and a 99% confidence interval for the argument
9.1 Program Text
Program Text (g07abce.c)
9.2 Program Data
Program Data (g07abce.d)
9.3 Program Results
Program Results (g07abce.r)