NAG Library Function Document

1Purpose

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

2Specification

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

3Description

Given a random sample of size $n$, denoted by ${x}_{1},{x}_{2},\dots ,{x}_{n}$, from a Poisson distribution with probability function
 $px=e-θ θxx! , x=0,1,2,…$
the point estimate, $\stackrel{^}{\theta }$, for $\theta$ is the sample mean, $\stackrel{-}{x}$.
Given $n$ and $\stackrel{-}{x}$ this function computes a $100\left(1-\alpha \right)%$ confidence interval for the argument $\theta$, denoted by [${\theta }_{l},{\theta }_{u}$], where $\alpha$ is in the interval $\left(0,1\right)$.
The lower and upper confidence limits are estimated by the solutions to the equations
 $e-nθl∑x=T∞ nθlxx! =α2, e-nθu∑x=0Tnθuxx! =α2,$
where $T=\sum _{i=1}^{n}{x}_{i}=n\stackrel{^}{\theta }$.
The relationship between the Poisson distribution and the ${\chi }^{2}$-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 ${\chi }_{\nu ,p}^{2}$ is the deviate associated with the lower tail probability $p$ of the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom.
In turn the relationship between the ${\chi }^{2}$-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 ${\gamma }_{\alpha ,\beta \text{;}\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the gamma distribution with shape argument $\alpha$ and scale argument $\beta$. 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

5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the sample size.
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{xmean}$doubleInput
On entry: the sample mean, $\stackrel{-}{x}$.
Constraint: ${\mathbf{xmean}}\ge 0.0$.
3:    $\mathbf{clevel}$doubleInput
On entry: the confidence level, $\left(1-\alpha \right)$, for two-sided interval estimate. For example ${\mathbf{clevel}}=0.95$ gives a $95%$ confidence interval.
Constraint: $0.0<{\mathbf{clevel}}<1.0$.
4:    $\mathbf{tl}$double *Output
On exit: the lower limit, ${\theta }_{l}$, of the confidence interval.
5:    $\mathbf{tu}$double *Output
On exit: the upper limit, ${\theta }_{u}$, of the confidence interval.
6:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
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.
NE_REAL
On entry, ${\mathbf{clevel}}\le 0.0$ or ${\mathbf{clevel}}\ge 1.0$: ${\mathbf{clevel}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{xmean}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{xmean}}\ge 0.0$.

7Accuracy

For most cases the results should have a relative accuracy of $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\text{0.5e}-12,50.0×\epsilon \right)$ where $\epsilon$ 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 $\alpha /2$ or $1-\alpha /2$ 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.

8Parallelism and Performance

nag_poisson_ci (g07abc) is not threaded in any implementation.

None.

10Example

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

10.1Program Text

Program Text (g07abce.c)

10.2Program Data

Program Data (g07abce.d)

10.3Program Results

Program Results (g07abce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017