10.3. Signed-Rank Wilcoxon 593We now derive some AREs between the Wilcoxon and thet-test. As noted
above, the parameterτW is a scale functional and, hence, varies directly with scale
transformations of the formaX,fora>0. Likewise, the standard deviationσis
also a scale functional. Therefore, because the AREs are ratios of scale functionals,
they are scale invariant. Hence, for derivations of AREs, we can select a pdf with a
convenient choice of scale. For example, if we are considering an ARE at the normal
distribution, we can work with theN(0,1) pdf.
Example 10.3.2(ARE(W, t) at the normal distribution).Iff(x)isaN(0,1) pdf,
then
τW−^1 =√
12∫∞−∞(
1
√
2 πexp{−x^2 / 2 }) 2
dx=√
12
√
2√
2 π∫∞−∞1
√
2 π(1/√
2)exp{− 2 −^1 (x/(1/√
2))^2 }dx=√
3
πHenceτW^2 =π/3. Sinceσ=1,wehave
ARE(W, t)=
σ^2
τW^2=
3
π=0. 955. (10.3.29)As discussed above, this ARE holds for all normal distributions. Hence, at the
normal distribution, the Wilcoxon signed-rank test is 95.5% efficient as thet-test.
The Wilcoxon is called ahighly efficientprocedure.
Example 10.3.3(ARE(W, t) at a Family of Contaminated Normals). For this
example, suppose thatf(x) is the pdf of a contaminated normal distribution. For
convenience, we use the standardized pdf given in expression (10.2.30) withb=1.
Recall that for this distribution, (1− ) proportion of the time the sample is drawn
from aN(0,1) distribution, while proportion of the time the sample is drawn from
aN(0,σc^2 ) distribution. Recall that the variance isσ^2 =1+ (σ^2 c−1). Note that
the formula for the pdff(x) is given in expression (3.4.17). In Exercise 10.3.5 it is
shown that ∫∞
−∞f^2 (x)dx=(1− )^2
2√
π+ 2
6√
π+(1− )
2√
π. (10.3.30)
Based on this, an expression for the ARE can be obtained; see Exercise 10.3.5. We
used this expression to determine the AREs between the Wilcoxon and thet-tests
for the situations withσc=3and varying from 0.00–0.25, displaying them in
Table 10.3.2. For convenience, we have also displayed the AREs between the sign
test and these two tests.
Note that the signed-rank Wilcoxon is more efficient than thet-test even at 1%
contamination and increases to 150% efficiency for 15% contamination.
10.3.2 Estimating Equations Based on Signed-Rank Wilcoxon
For the sign procedure, the estimation ofθwas based on minimizing theL 1 norm.
The estimator associated with the signed-rank test minimizes another norm, which