Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.3. Signed-Rank Wilcoxon 591

Letcαdenote the critical value of a levelαtest of the hypotheses (10.3.1) based
on the signed-rank test statistic T+; i.e.,α =PH 0 (T+≥cα). LetγSW(θ)=
Pθ(T+≥cα), forθ≥θ 0 , denote the power function of the test. The translation
property, Lemma 10.2.1, holds for the signed-rank Wilcoxon. Hence, as in Theorem
10.2.1, the power function is a nondecreasing function ofθ. In particular, the
signed-rank Wilcoxon test is an unbiased test for the one-sided hypotheses (10.3.1).

10.3.1 AsymptoticRelativeEfficiency

We investigate the efficiency of the signed-rank Wilcoxon by first determining its

efficacy. Without loss of generality, we can assume thatθ 0 = 0. Consider the same

sequence of local alternatives discussed in the last section; i.e.,

H 0 : θ=0versusH 1 n:θn=√δn, (10.3.16)

whereδ>0. Contemplate the modified statistic, which is the average ofT+(θ),

T

+

(θ)=

2

n(n+1)

T+(θ). (10.3.17)

Then, by (10.3.12),

E 0 [T

+

(0)] =n(n^2 +1)n(n 4 +1)=^12 and σ^2 T+(0) = Var 0 [T

+

(0)] = 6 n^2 (nn+1+1). (10.3.18)

Letan=2/n(n+ 1). Note that we can decomposeT

• (θn)intotwopartsas

T

+

(θn)=anS(θn)+an

∑

i<j

I(Xi+Xj> 2 θn)=anS(θn)+anT∗(θn), (10.3.19)

whereS(θ) is the sign process (10.2.9) and

T∗(θn)=

∑

i<j

I(Xi+Xj> 2 θn). (10.3.20)

To obtain the efficacy, we require the mean

μT+(θn)=Eθn[T

+

(0)] =E 0 [T

+

(−θn)]. (10.3.21)

But by (10.2.14),anE 0 (S(−θn)) =ann(2−^1 −F(−θn))→0. Henceweneedonly
be concerned with the second term in (10.3.19). But note that the Walsh averages
inT∗(θ) are identically distributed. Thus

anE 0 (T∗(−θn)) =an

(

n

2

)

P 0 (X 1 +X 2 >− 2 θn). (10.3.22)