10.4. Mann–Whitney–Wilcoxon Procedure 603where we have applied the mean value theorem to obtain the last line. Putting
together (10.4.17) and (10.4.19), we have the efficacycU= lim
n→∞μ′U(0)
√
nσU(0)=√
12√
λ 1 λ 2∫∞−∞f^2 (x)dx. (10.4.20)This derivation can be made rigorous, leading to the following theorem:
Theorem 10.4.2(Asymptotic Power Lemma).Consider the sequence of hypotheses
(10.4.14). The limit of the power function of the sizeαMann–Whitney–Wilcoxon
test is given by
lim
n→∞
γU(Δn)=1−Φ(
zα−√
λ 1 λ 2 δτW−^1)
, (10.4.21)whereτW=1/
√
12∫∞
−∞f(^2) (x)dxis the reciprocal of the efficacyc
UandΦ(z)is the
cdf of a standard normal random variable.
As in the last two sections, we can use this theorem to establish a relative mea-
sure of efficiency by considering sample size determination. Consider the hypotheses
(10.4.4). Suppose we want to determine the sample sizen=n 1 +n 2 for a levelα
MWW test to detect the alternative Δ∗with approximate powerγ∗.ByTheorem
10.4.2, we have the equation
γ∗=γU(
√
nΔ∗/
√
n)≈ 1 −Φ(zα−
√
λ 1 λ 2
√
nΔ∗τW−^1 ). (10.4.22)
This leads to the equation
zγ∗=zα−
√
λ 1 λ 2 δτW−^1 , (10.4.23)
where Φ(zγ∗)=1−γ∗.Solvingforn,weobtain
nU≈
(
(zα−zγ∗)τW
Δ∗
√
λ 1 λ 2
) 2
. (10.4.24)
To use this in applications, the sample size proportionsλ 1 =n 1 /nandλ 2 =n 2 /n
must be given. As Exercise 10.4.1 points out, the most powerful two-sample designs
have sample size proportions of 1/2, i.e., equal sample sizes.
To use this to obtain the asymptotic relative efficiency between the MWW and
the two-sample pooledt-test, Exercise 10.4.2 shows that the sample size needed for
the two-samplet-tests to attain approximate powerγ∗to detect Δ∗is given by
nLS≈(
(zα−zγ∗)σ
Δ∗√
λ 1 λ 2) 2
, (10.4.25)whereσis the variance ofei. Hence, as in the last section, the asymptotic relative
efficiency between the Wilcoxon test (MWW) and thet-test is the ratio of the
sample sizes (10.4.24) and (10.4.25), which is
ARE(MWW,LS) =
σ^2
τW^2. (10.4.26)