7.2. A Sufficient Statistic for a Parameter 419he knows they will probably not equal the originalx-values) as follows. He
notes that the conditional probability of independent Poisson random vari-
ablesZ 1 ,Z 2 ,...,Znbeing equal toz 1 ,z 2 ,...,zn,given∑
zi=y 1 ,isθz^1 e−θ
z 1!θz^2 e−θ
z 2! ···θzne−θ
zn!
(nθ)y^1 e−nθ
y 1!=y 1!
z 1 !z 2 !···zn!(
1
n)z 1 (
1
n)z 2
···(
1
n)znsinceY 1 =∑
Zihas a Poisson distribution with meannθ. The latter distri-
bution is multinomial withy 1 independent trials, each terminating in one ofn
mutually exclusive and exhaustive ways, each of which has the same probabil-
ity 1/n. Accordingly,Bruns such a multinomial experimenty 1 independent
trials and obtainsz 1 ,z 2 ,...,zn. Find the likelihood function using thesez-
values. Is it proportional to that of statisticianA?
Hint:Here the likelihood function is the product of this conditional pdf and
the pdf ofY 1 =∑
Zi.7.2 ASufficientStatisticforaParameter..................
Suppose thatX 1 ,X 2 ,...,Xnis a random sample from a distribution that has pdf
f(x;θ),θ∈Ω. In Chapters 4 and 6, we constructed statistics to make statistical
inferences as illustrated by point and interval estimation and tests of statistical
hypotheses. We note that a statistic, for example,Y=u(X 1 ,X 2 ,...,Xn), is a form
of data reduction. To illustrate, instead of listing all of the individual observations
X 1 ,X 2 ,...,Xn, we might prefer to give only the sample meanXor the sample
varianceS^2. Thus statisticians look for ways of reducing a set of data so that these
data can be more easily understood without losing the meaning associated with the
entire set of observations.
It is interesting to note that a statisticY=u(X 1 ,X 2 ,...,Xn) really partitions
the sample space ofX 1 ,X 2 ,...,Xn. For illustration, suppose we say that the sample
was observed andx=8.32. There are many points in the sample space which
have that same mean of 8.32, and we can consider them as belonging to the set
{(x 1 ,x 2 ,...,xn):x=8. 32 }. As a matter of fact, all points on the hyperplane
x 1 +x 2 +···+xn=(8.32)nyield the mean ofx=8.32, so this hyperplane is the set. However, there are many
values thatXcan take, and thus there are many such sets. So, in this sense, the
sample meanX,oranystatisticY =u(X 1 ,X 2 ,...,Xn), partitions the sample
space into a collection of sets.
Often in the study of statistics the parameterθof the model is unknown; thus,
we need to make some statistical inference about it. In this section we consider a
statistic denoted byY 1 =u 1 (X 1 ,X 2 ,...,Xn), which we call asufficient statistic
and which we find is good for making those inferences. This sufficient statistic
partitions the sample space in such a way that, given
(X 1 ,X 2 ,...,Xn)∈{(x 1 ,x 2 ,...,xn):u 1 (x 1 ,x 2 ,...,xn)=y 1 },