naginterfaces.library.univar.ci_​poisson

naginterfaces.library.univar.ci_poisson(n, xmean, clevel)

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

For full information please refer to the NAG Library document for g07ab

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g07/g07abf.html

Parameters

nint

$n$, the sample size.

xmeanfloat

The sample mean, $\overline{x}$.

clevelfloat

The confidence level, $(1-\alpha )$, for two-sided interval estimate. For example $clevel=0.95$ gives a $95\%$ confidence interval.

Returns

tlfloat

The lower limit, ${\theta }_l$, of the confidence interval.

tufloat

The upper limit, ${\theta }_u$, of the confidence interval.

Raises

NagValueError

(errno $1$)

On entry, $clevel=⟨value⟩$.

Constraint: $0.0<clevel<1.0$.

(errno $1$)

On entry, $xmean=⟨value⟩$.

Constraint: $xmean\ge 0.0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $2$)

When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge.

Notes

Given a random sample of size $n$, denoted by $x_1,x_2,\dots ,x_n$, from a Poisson distribution with probability function

$$p\left(x\right)=e^{-\theta }\frac{{\theta }^x}{x!}\text{,}x=0,1,2,\dots$$

the point estimate, $\hat{\theta }$, for $\theta$ is the sample mean, $\overline{x}$.

Given $n$ and $\overline{x}$ this function computes a $100(1-\alpha )\%$ confidence interval for the parameter $\theta$, denoted by [${\theta }_l,{\theta }_u$], where $\alpha$ is in the interval $(0,1)$.

The lower and upper confidence limits are estimated by the solutions to the equations

$$\begin{array}{c}\begin{array}{c}{e}^{-n{\theta }_l}{\sum }_{x=T}^{\infty }\frac{{\left(n{\theta }_l\right)}^x}{x!}=\frac{\alpha }{2}\text{,}\\ \\ \\ e^{-n{\theta }_u}{\sum }_{x=0}^T\frac{{\left(n{\theta }_u\right)}^x}{x!}=\frac{\alpha }{2}\text{,}\end{array}\end{array}$$

where $T={\sum }_{i=1}^n{x}_i=n\hat{\theta }$.

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

$$\begin{array}{c}\begin{array}{c}{\theta }_l=\frac{1}{2n}{\chi }_{2T,\alpha /2}^2\text{,}\\ \\ {\theta }_u=\frac{1}{2n}{\chi }_{2T+2,1-\alpha /2}^2\text{,}\end{array}\end{array}$$

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;

$$\begin{array}{c}\begin{array}{c}{\theta }_l=\frac{1}{2n}{\gamma }_{T,2\text{;}\alpha /2}\text{,}\\ \\ {\theta }_u=\frac{1}{2n}{\gamma }_{T+1,2\text{;}1-\alpha /2}\text{,}\end{array}\end{array}$$

where ${\gamma }_{\alpha ,\beta \text{;}\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$. These deviates are computed using stat.inv_cdf_gamma.

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