10.2. Sample Median and the Sign Test 577Recall from Definition 4.5.4 of Chapter 4 that the size of the test for a composite
null hypothesis is given by maxθ≤θ 0 γ(θ). Becauseγ(θ) is nondecreasing, the size of
the sign test isαfor this extended null hypothesis. As a second result, it follows
immediately that the sign test is an unbiased test; see Section 8.3. As Exercise
10.2.8 shows, the power function of the sign test for the other one-sided alternative,
H 1 :θ<θ 0 , is nonincreasing.
Under an alternative, sayθ =θ 1 , the test statisticS(θ 0 ) has the binomial
distributionb(n, p 1 ), wherep 1 is given byp 1 =Pθ 1 (X>0) = 1−F(−θ 1 ), (10.2.12)whereF(x)isthecdfofεin Model (10.2.1). HenceS(θ 0 ) is not distribution free
under alternative hypotheses. As in Exercise 10.2.3, we can determine the power of
the test for specifiedθ 1 andF(x). We want to compare the power of the sign test
to other sizeαtests, in particular the test based on the sample mean. However,
for these comparison purposes, we need more general results, some of which are
obtained in the next subsection.
10.2.1 AsymptoticRelativeEfficiency
One solution to this problem is to consider the behavior of a test under a sequence
of local alternatives. In this section, we often takeθ 0 = 0 in hypotheses (10.2.2). As
noted before Lemma 10.2.1, this is without loss of generality. For the hypotheses
(10.2.2), consider the sequence of alternatives
H 0 : θ=0versusH 1 n:θn=√δn, (10.2.13)whereδ>0. Note that this sequence of alternatives converges to the null hypothesis
asn→∞. We often call such a sequence of alternativeslocal alternatives.The
idea is to consider how the power function of a test behaves relative to the power
functions of other tests under this sequence of alternatives. We only sketch this
development. For more details, the reader can consult the more advanced books
cited in Section 10.1. As a first step in that direction, we obtain the asymptotic
power lemma for the sign test.
Consider the large sample sizeαtest given by (10.2.6). Under the alternative