Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.5.∗Minimax and Classification Procedures 507

8.4.2.LetXhave a Poisson distribution with meanθ. Find the sequential proba-

bility ratio test for testingH 0 :θ=0.02 againstH 1 :θ=0.07. Show that this test

can be based upon the statistic

∑n
1 Xi.Ifαa=0.20 andβa=0.10, findc^0 (n)and
c 1 (n).

8.4.3.Let the independent random variablesY andZbeN(μ 1 ,1) andN(μ 2 ,1),
respectively. Letθ= μ 1 −μ 2. Let us observe independent observations from
each distribution, sayY 1 ,Y 2 ,...andZ 1 ,Z 2 ,.... To test sequentially the hypothesis
H 0 :θ= 0 againstH 1 :θ=^12 , use the sequenceXi=Yi−Zi,i=1, 2 ,....If

αa=βa=0.05, show that the test can be based uponX=Y−Z.Findc 0 (n)and
c 1 (n).

8.4.4. Suppose that a manufacturing process makes about 3% defective items,
which is considered satisfactory for this particular product. The managers would
like to decrease this to about 1% and clearly want to guard against a substantial
increase, say to 5%. To monitor the process, periodicallyn= 100 items are taken
and the numberX of defectives counted. Assume thatXisb(n= 100,p=θ).
BasedonasequenceX 1 ,X 2 ,...,Xm,..., determine a sequential probability ratio
test that testsH 0 :θ=0.01 againstH 1 :θ=0.05. (Note thatθ=0.03, the present
level, is in between these two values.) Write this test in the form

h 0 >

∑m

i=1

(xi−nd)>h 1

and determined,h 0 ,andh 1 ifαa=βa=0.02.

8.4.5.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution with pdff(x;θ)=
θxθ−^1 , 0 <x<1, zero elsewhere.

(a)Find a complete sufficient statistic forθ.

(b)Ifαa=βa= 101 , find the sequential probability ratio test ofH 0 :θ= 2 against

H 1 :θ=3.

8.5 ∗MinimaxandClassificationProcedures

We have considered several procedures that may be used in problems of point es-
timation. Among these were decision function procedures (in particular, minimax
decisions). In this section, we apply minimax procedures to the problem of testing a
simple hypothesisH 0 against an alternative simple hypothesisH 1 .Itisimportant
to observe that these procedures yield, in accordance with the Neyman–Pearson
theorem, a best test ofH 0 againstH 1. We end this section with a discussion on an
application of these procedures to a classification problem.

8.5.1 MinimaxProcedures.......................

We first investigate the decision function approach to the problem of testing a simple

null hypothesis against a simple alternative hypothesis. Let the joint pdf of then