Page 293 - Fundamentals of Probability and Statistics for Engineers

276 Fundamentals of Probability and Statistics for Engineers

The foregoing results can be extended to the multiple parameter case. Let
T
q ˆ [ 1 ... m ], m n, be the parameter vector. Then Y 1 ˆ h 1 (X 1 ,..., X n ), ...,
Y r ˆ h r X 1 , .. . , X n ), r m, is a set of sufficient statistics for if and only ifq
n
Y
f X x j ; q†ˆ g 1 ‰h x 1 ; ... ; x n †; qŠg 2 x 1 ; ... ; x n †; 9:51†
jˆ1
T
where h ˆ [h 1 h r ]. A similar expression holds when X is discrete.
Example 9.7. Let us show that statistic X is a sufficient statistic for in
Example 9.5. In this case,

n n
Y Y
p X x j ; †ˆ 1 † 1 x j
x j
jˆ1 jˆ1 9:52†
ˆ x j 1 † n x j :

We see that the joint probability mass function (jpmf) is a function of x j and
.If we let
n
X
Y ˆ X j ;
jˆ1
the jpmf of X 1 ,... , and X n takes the form given by Equation (9.50), with

g 1 ˆ x j 1 † n x j ;

and
g 2 ˆ 1:

In this example,
n
X
X j
jˆ1
^

is thus a sufficient statistic for . We have seen in Example 9.5 that both 1 and
^
^
^
2 , where ˆ X , and ˆ nX ‡ 1)/ n ‡ 2), are based on this sufficient
1
2
^
statistic. Furthermore, 1 , being unbiased, is a sufficient estimator for .

Example 9.8. Suppose X 1 , X 2 ,... , and X n are a sample taken from a Poisson

distribution; that is,
k
e
p X k; †ˆ ; k ˆ 0; 1; 2; ... ; 9:53†
k!
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