Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

484 Optimal Tests of Hypotheses

Example 8.2.5.LetX 1 ,X 2 ,...,Xnbe a random sample from a Bernoulli distri-
bution with parameterp=θ,where0<θ<1. Letθ′<θ′′. Consider the ratio of
likelihoods

L(θ′;x 1 ,x 2 ,...,xn)

L(θ′′;x 1 ,x 2 ,...,xn)

=

(θ′)

Px

i(1−θ′)n−

Px

i

(θ′′)

P

xi(1−θ′′)n−

P

xi=

[

θ′(1−θ′′)

θ′′(1−θ′)

]Pxi(

1 −θ′

1 −θ′′

)n

.

Sinceθ′/θ′′<1and(1−θ′′)/(1−θ′)<1, so thatθ′(1−θ′′)/θ′′(1−θ′)<1, the
ratio is a decreasing function ofy=

∑

xi. Thus we have a monotone likelihood

ratio in the statisticY=

∑

Xi.

Consider the hypotheses

H 0 :θ≤θ′versusH 1 : θ>θ′. (8.2.4)

By our discussion above, the UMP levelαdecision rule for testingH 0 versusH 1 is
given by

RejectH 0 ifY=

∑n

i=1Xi≥c,

wherecis such thatα=Pθ′[Y≥c].

In the last example concerning a Bernoulli pmf, we obtained a UMP test by
showing that its likelihood possesses mlr. The Bernoulli distribution is a regular
case of the exponential family and our argument, under the one assumption below,
can be generalized to the entire regular exponential family. To show this, suppose
that the random sampleX 1 ,X 2 ,...,Xnarises from a pdf or pmf representing a
regular case of the exponential class, namely,

f(x;θ)=

{

exp[p(θ)K(x)+H(x)+q(θ)] x∈S

0elsewhere,

where the support ofX,S,isfreeofθ. Further assume thatp(θ)isanincreasing
function ofθ.Then

L(θ′)

L(θ′′)

=

exp

[

p(θ′)

∑n

1

K(xi)+

∑n

1

H(xi)+nq(θ′)

]

exp

[

p(θ′′)

∑n

1

K(xi)+

∑n

1

H(xi)+nq(θ′′)

]

=exp

{

[p(θ′)−p(θ′′)]

∑n

1

K(xi)+n[q(θ′)−q(θ′′)]

}

.

Ifθ′<θ′′,p(θ) being an increasing function, requires this ratio to be a decreasing
function ofy=

∑n
1 K(xi). Thus, we have a monotone likelihood ratio in the statistic
Y=

∑n

1 K(Xi). Hence consider the hypotheses

H 0 :θ≤θ′versusH 1 : θ>θ′. (8.2.5)