Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

502 Optimal Tests of Hypotheses

A test of this kind is called Wald’ssequential probability ratio test.Fre-
quently, inequality (8.4.3) can be conveniently expressed in an equivalent form:

c 0 (n)<u(x 1 ,x 2 ,...,xn)<c 1 (n), (8.4.4)

whereu(X 1 ,X 2 ,...,Xn)isastatisticandc 0 (n)andc 1 (n) depend on the constants
k 0 ,k 1 ,θ′,θ′′,andonn. Then the observations are stopped and a decision is reached
as soon as

u(x 1 ,x 2 ,...,xn)≤c 0 (n)oru(x 1 ,x 2 ,...,xn)≥c 1 (n).

We now give an illustrative example.

Example 8.4.1.LetXhave a pmf

f(x;θ)=

{

θx(1−θ)^1 −x x=0, 1

0elsewhere.

In the preceding discussion of a sequential probability ratio test, letH 0 :θ=^13 and
H 1 :θ=^23 ; then, with

∑

xi=

∑n

i=1xi,

L(^13 ,n)

L(^23 ,n)

=

(^13 )

Px

i(^2

3 )

n−Pxi

(^23 )

Px

i(^1

3 )

n−Pxi=2

n− 2

P

xi.

If we take logarithms to the base 2, the inequality

k 0 <

L(^13 ,n)

L(^23 ,n)

<k 1 ,

with 0<k 0 <k 1 , becomes

log 2 k 0 <n− 2

∑n

1

xi<log 2 k 1 ,

or, equivalently, in the notation of expression (8.4.4),

c 0 (n)=

n

2

−

1

2

log 2 k 1 <

∑n

1

xi<

n

2

−

1

2

log 2 k 0 =c 1 (n).

Note thatL(^13 ,n)/L(^23 ,n)≤k 0 if and only ifc 1 (n)≤

∑n

1 xi;andL(

1

3 ,n)/L(

2
3 ,n)≥
k 1 if and only ifc 0 (n)≥

∑n

1 xi. Thus we continue to observe outcomes as long as

c 0 (n)<

∑n
1 xi<c^1 (n). The observation of outcomes is discontinued with the first
value ofnofNfor which eitherc 1 (n)≤

∑n

1 xiorc^0 (n)≥

∑n
1 xi. The inequality
c 1 (n)≤

∑n

1 xileads to rejection ofH^0 :θ=

1

3 (the acceptance ofH^1 ), and the

inequalityc 0 (n)≥

∑n

1 xileads to the acceptance ofH^0 :θ=

1

3 (the rejection of

H 1 ).

Remark 8.4.1.At this point, the reader undoubtedly sees that there are many

questions that should be raised in connection with the sequential probability ratio

test. Some of these questions are possibly among the following: