Fundamentals of Probability and Statistics for Engineers

The foregoing results can be extended to the multiple parameter case. Let

be the parameter vector. Then Y 1 h 1 (X 1 ,...,Xn),...,

m, is a set of sufficient statistics for if and only if

where hT A similar expression holds when X is discrete.

Example 9.7.Let us show that statisticX is a sufficient statistic for in
Example 9.5. In this case,

We see that the joint probability mass function (jpmf) is a function of and
.Ifwelet

the jpmf of X 1 ,...,and Xn takes the form given by Equation (9.50), with

and

In this example,

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

Ex ample 9. 8. Suppose X 1 ,X 2 ,...,andXn are a sample taken from a Poisson
distribution; that is,

276 Fundamentals of Probability and Statistics for Engineers

qTˆ[ 1 ...m],mn, ˆ
YrˆhrX 1 ,...,Xn),r q

Yn

jˆ 1

fX…xj;q†ˆg 1 ‰h…x 1 ;...;xn†;qŠg 2 …x 1 ;...;xn†; … 9 : 51 †

ˆ[h 1 hr].



Yn

jˆ 1

pX…xj;†ˆ

Yn

jˆ 1

xj… 1 †^1 xj

ˆxj… 1 †nxj:

… 9 : 52 †

xj


Yˆ

Xn

jˆ 1

Xj;

g 1 ˆxj… 1 †nxj;

g 2 ˆ 1 :

Xn

jˆ 1

Xj

^

^ ^ ˆ ^ ˆnX‡1)/n‡2),
^ 



pX…k;†ˆ

ke

k!

; kˆ 0 ; 1 ; 2 ;...; … 9 : 53 †