Open Journal of Statistics, 2012, 2, 204-207
http://dx.doi.org/10.4236/ojs.2012.22024 Published Online April 2012 (http://www.SciRP.org/journal/ojs)
Approximate Confidence Interval for the Mean of
Poisson Distribution
Manad Khamkong
Department of Statistics, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand
Email: manad.k@cmu.ac.th
Received February 19, 2012; revised March 20, 2012; accepted April 5, 2012
ABSTRACT
A Poisson distribution is well used as a standard model for analyzing count data. Most of the usual constructing confi-
dence intervals are based on an asymptotic approximation to the distribution of the sample mean by using the Wald in-
terval. That is, the Wald interval has poor performance in terms of coverage probabilities and average widths interval
for small means and small to moderate sample sizes. In this paper, an approximate confidence interval for a Poisson
mean is proposed and is based on an empirically determined the tail probabilities. Simulation results show that the pro-
posed interval outperforms the others when small means and small to moderate sample sizes.
Keywords: Confidence Interval; Coverage Probability; Poisson Distribution; Expected Width; Wald Interval
1. Introduction
In many applications, the variable of interest is given in
the form of an event count or a non-negative integer
value which refers to the number of a occurrences of
particular phenomenon over a fixed set of time, distance,
area or space. Some examples of such data are number of
road accident victims per week, number of cases with a
specific disease in epidemiology, etc. Poisson distribu-
tion is a standard and good model for analyzing count
data and it seems to be the most common and frequently
used as well.
It is very interesting to construct a confidence interval
for a Poisson mean. Suppose
1 2 n
is a random
sample of size n from a Poisson (
X ,X , ,X
0 ) distribution. A
problem in finding an exact 1 two-sided confidence
interval for mean ( ,U X LX
) of Poissonity is
given by
L
U
2
,
2
LX PX x
UX PX x
(1)
where
L
and
U
are, respectively, the lower and up-
per endpoints of the confidence interval.
Let
n
1
i
i1
X n X
ˆ
is the maximum likelihood es-
timator of . As n large by central limit theorem, the
Wald interval for the mean is given by
2
X
X z
n
, (2)
where
2
z
is the ( 1 2 )100
th
percentile of the stan-
dard normal distribution. The Wald interval with conti-
nuity correction interval (WCC) uses a normal distribu-
tion to approximate a Poisson distribution is defined as
2
X 0.5
X z
n
, (3)
Several methods have been proposed to construct a
confidence interval for a Poisson mean such as Cai [1],
Byrne and Kabaila [2], Guan [3], Krishnamoorthy and
Peng [4], Stamey and Hamillton [5], Swifi [6] and others.
Guan [3] has suggested that the score interval (SC) is the
uppermost approximation on interval estimation of a
Poisson mean for moderate is given by
2
2
2
2
2
z
X
z
4n
X z
2n n
(4)
and he has also proposed the moved score confidence
interval (MSC) as follows,
2
2
2
2
2
z
X
0.46z
4n
X z
n n
(5)
Barker [7] has recommended the exact confidence in-
terval outperform but not explicit closed form and was
computed difficult. In particular, the Wald interval with
continuity correction interval (WCC) achieves coverage
probabilities quite faster than the Wald interval. However,
The WCC is known to perform poorly for small to mod-
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