Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.1. Measures of Quality of Estimators 417

Let us observe, however, that one of the corresponding estimators,Y/nand 1/Z,
is biased. We have

E

(

Y

10

)

=

1

10

E(Y)=

1

10

(10θ)=θ,

while

E

(

1

Z

)

=

∑∞

z=1

1

z

(1−θ)z−^1 θ

= θ+^12 (1−θ)θ+^13 (1−θ)^2 θ+···>θ.

That is, 1/Zis a biased estimator whileY/10 is unbiased. ThusAis using an
unbiased estimator whileBis not. Should we adjustB’s estimator so that it, too,
is unbiased?
It is interesting to note that if we maximize the two respective likelihood func-
tions, namely,

L 1 (θ)=

(

10

y

)

θy(1−θ)^10 −y

and

L 2 (θ)=(1−θ)z−^1 θ,

withn=10,y=1,andz= 10, we get exactly the same answer,θˆ= 101 .This
must be the case, because in each situation we are maximizing (1−θ)^9 θ.Many
statisticians believe that this is the way it should be and accordingly adopt the
likelihood principle:
Suppose two different sets of data from possibly two different random experiments
lead to respective likelihood ratios,L 1 (θ)andL 2 (θ), that are proportional to each
other. These two data sets provide the same information about the parameterθand
a statistician should obtain the same estimate ofθfrom either.
In our special illustration, we note thatL 1 (θ)∝L 2 (θ), and the likelihood princi-
ple states that statisticiansAandBshould make the same inference. Thus believers
in the likelihood principle would not adjust the second estimator to make it unbi-
ased.

EXERCISES

7.1.1. Show that the meanXof a random sample of sizenfrom a distribution
having pdff(x;θ)=(1/θ)e−(x/θ), 0 <x<∞, 0 <θ<∞, zero elsewhere, is an
unbiased estimator ofθand has varianceθ^2 /n.

7.1.2. LetX 1 ,X 2 ,...,Xndenote a random sample from a normal distribution
with mean zero and varianceθ, 0 <θ<∞. Show that

∑n

1 X

2

i/nis an unbiased

estimator ofθand has variance 2θ^2 /n.