Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.2. Sample Median and the Sign Test 579

mean given in (10.2.14) in terms of another concept calledefficacy.Consider

another standardization of the test statistic given by

S(0) =

1

n

∑n

i=1

I(Xi>0), (10.2.16)

where the bar notation is used to signify thatS(0) is an average ofI(Xi>0) and,
in this case underH 0 , converges in probability to^12 .Letμ(θ)=Eθ(S(0)−^12 ).
Then, by expression (10.2.14), we have

μ(θn)=Eθn

(

S(0)−

1

2

)

=

1

2

−F(−θn). (10.2.17)

Letσ^2 S=Var(S(0)) = 41 n. Finally, define theefficacyof the sign test to be

cS= lim

n→∞

μ′(0)

√

nσS

. (10.2.18)

That is, the efficacy is the rate of change of the mean of the test statistic at the
null divided by the product of

√

nand the standard deviation of the test statistic

at the null. So the efficacy increases with an increase in this rate, as it should. We

use this formulation of efficacy throughout this chapter.

Hence, by expression (10.2.14), the efficacy of the sign test is

cS=

f(0)

1 / 2

=2f(0) =τS−^1 , (10.2.19)

the reciprocal of the scale parameterτS. In terms of efficacy, we can write the
conclusion of the Asymptotic Power Lemma as

lim

n→∞

γ(θn)=1−Φ(zα−δcS). (10.2.20)

This is not a coincidence, and it is true for the procedures we consider in the next
section.

Remark 10.2.1.In this chapter, we compare nonparametric procedures with tra-
ditional parametric procedures. For instance, we compare the sign test with the test
based on the sample mean. Traditionally, tests based on sample means are referred
to ast-tests. Even though our comparisons are asymptotic and we could use the
terminology ofz-tests, we instead use the traditional terminology oft-tests.

As a second illustration of efficacy, we determine the efficacy of thet-test for the
mean. Assume that the random variablesεiin Model (10.2.1) are symmetrically
distributed about 0 and their mean exists. Hence the parameterθis the location
parameter. In particular,θ=E(Xi)=med(Xi). Denote the variance ofXibyσ^2.
This allows us to easily compare the sign andt-tests. Recall for hypotheses (10.2.2)
that thet-test rejectsH 0 in favor ofH 1 ifX≥c. The form of the test statistic is
thenX.Furthermore,wehave

μX(θ)=Eθ(X)=θ (10.2.21)