Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.2. Uniformly Most Powerful Tests 485

By our discussion above concerning mlr, the UMP levelαdecision rule for testing

H 0 versusH 1 is given by

RejectH 0 ifY=

∑n

i=1

K(Xi)≥c,

wherecis such thatα=Pθ′[Y≥c]. Furthermore, the power function of this test
is an increasing function inθ.
For the record, consider the other one-sided alternative hypotheses,

H 0 :θ≥θ′versusH 1 : θ<θ′. (8.2.6)

The UMP levelαdecision rule is, forp(θ) an increasing function,

RejectH 0 ifY=

∑n

i=1

K(Xi)≤c,

wherecis such thatα=Pθ′[Y≤c].
If in the preceding situation with monotone likelihood ratio we testH 0 :θ=
θ′ againstH 1 :θ>θ′,then

∑
K(xi)≥cwould be a uniformly most powerful
critical region. From the likelihood ratios displayed in Examples 8.2.2–8.2.5, we see
immediately that the respective critical regions

∑n

i=1

x^2 i≥c,

∑n

i=1

xi≥c,

∑n

i=1

xi≥c,

∑n

i=1

xi≥c

are uniformly most powerful for testingH 0 :θ=θ′againstH 1 :θ>θ′.
There is a final remark that should be made about uniformly most powerful
tests. Of course, in Definition 8.2.1, the worduniformlyis associated withθ;that
is,Cis a best critical region of sizeαfor testingH 0 :θ=θ 0 against allθvalues
given by the composite alternativeH 1. However, suppose that the form of such a
region is
u(x 1 ,x 2 ,...,xn)≤c.

Then this form provides uniformly most powerful critical regions for all attainableα
values by, of course, appropriately changing the value ofc. That is, there is a certain
uniformity property, also associated withα, that is not always noted in statistics
texts.

EXERCISES

8.2.1. LetXhave the pmff(x;θ)=θx(1−θ)^1 −x,x=0,1, zero elsewhere. We
test the simple hypothesisH 0 :θ=^14 against the alternative composite hypothesis
H 1 :θ<^14 by taking a random sample of size 10 and rejectingH 0 :θ=^14 if and

∑only if the observed valuesx^1 ,x^2 ,...,x^10 of the sample observations are such that
10
1 xi≤1. Find the power functionγ(θ),^0 <θ≤

1

4 ,ofthistest.