nag_poisson_ci (g07abc) (PDF version)
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NAG Library Manual

# NAG Library Function Documentnag_poisson_ci (g07abc)

## 1  Purpose

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

## 2  Specification

 #include #include
 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
 $px=e-θ θxx! , x=0,1,2,…$
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).

## 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:     nIntegerInput
On entry: $n$, the sample size.
Constraint: ${\mathbf{n}}\ge 1$.
2:     xmeandoubleInput
On entry: the sample mean, $\stackrel{-}{x}$.
Constraint: ${\mathbf{xmean}}\ge 0.0$.
3:     cleveldoubleInput
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:     tldouble *Output
On exit: the lower limit, ${\theta }_{l}$, of the confidence interval.
5:     tudouble *Output
On exit: the upper limit, ${\theta }_{u}$, of the confidence interval.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_BAD_PARAM
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.
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: ${\mathbf{xmean}}\ge 0.0$.

## 7  Accuracy

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.

Not applicable.

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 $98$ 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 $\theta$. nag_poisson_ci (g07abc) is then called to compute both a 95% and a 99% confidence interval for the argument $\theta$.

### 10.1  Program Text

Program Text (g07abce.c)

### 10.2  Program Data

Program Data (g07abce.d)

### 10.3  Program Results

Program Results (g07abce.r)

nag_poisson_ci (g07abc) (PDF version)
g07 Chapter Contents
g07 Chapter Introduction
NAG Library Manual