Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

420 Sufficiency

the conditional probability ofX 1 ,X 2 ,...,Xndoes not depend uponθ.Intuitively,
this means that once the set determined byY 1 =y 1 is fixed, the distribution of
another statistic, sayY 2 =u 2 (X 1 ,X 2 ,...,Xn), does not depend upon the parameter
θbecause the conditional distribution ofX 1 ,X 2 ,...,Xndoes not depend uponθ.
Hence it is impossible to useY 2 ,givenY 1 =y 1 , to make a statistical inference about
θ.So,inasense,Y 1 exhaustsall the information aboutθthat is contained in the
sample. This is why we callY 1 =u 1 (X 1 ,X 2 ,...,Xn) a sufficient statistic.
To understand clearly the definition of a sufficient statistic for a parameterθ,
we start with an illustration.

Example 7.2.1.LetX 1 ,X 2 ,...,Xndenote a random sample from the distribution
that has pmf

f(x;θ)=

{

θx(1−θ)^1 −x x=0,1; 0<θ< 1

0elsewhere.

The statisticY 1 =X 1 +X 2 +···+Xnhas the pmf

fY 1 (y 1 ;θ)=

{(n

y 1

)

θy^1 (1−θ)n−y^1 y 1 =0, 1 ,...,n

0elsewhere.

What is the conditional probability

P(X 1 =x 1 ,X 2 =x 2 ,...,Xn=xn|Y 1 =y 1 )=P(A|B),

say, wherey 1 =0, 1 , 2 ,...,n? Unless the sum of the integersx 1 ,x 2 ,...,xn(each of
which equals zero or 1) is equal toy 1 , the conditional probability obviously equals
zero becauseA∩B=φ.Butinthecasey 1 =

∑

xi,wehavethatA⊂B,sothat

A∩B=AandP(A|B)=P(A)/P(B); thus, the conditional probability equals

θx^1 (1−θ)^1 −x^1 θx^2 (1−θ)^1 −x^2 ···θxn(1−θ)^1 −xn

(

n

y 1

)

θy^1 (1−θ)n−y^1

=

θ

Px

i(1−θ)n−

Px

i

(

n

∑

xi

)

θ

P

xi(1−θ)n−

P

xi

=

1

(

n

∑

xi

).

Sincey 1 =x 1 +x 2 +···+xnequals the number of ones in thenindependent trials,
this is the conditional probability of selecting a particular arrangement ofy 1 ones
and (n−y 1 ) zeros. Note that this conditional probability doesnotdepend upon
the value of the parameterθ.

In general, letfY 1 (y 1 ;θ) be the pmf of the statisticY 1 =u 1 (X 1 ,X 2 ,...,Xn),
whereX 1 ,X 2 ,...,Xnis a random sample arising from a distribution of the discrete
type having pmff(x;θ),θ∈Ω. The conditional probability ofX 1 =x 1 ,X 2 =
x 2 ,...,Xn=xn,givenY 1 =y 1 ,equals

f(x 1 ;θ)f(x 2 ;θ)···f(xn;θ)

fY 1 [u 1 (x 1 ,x 2 ,...,xn);θ]

,