606 Nonparametric and Robust Statisticsindwil <-0; indt <- 0
for(i in 1:nsims){
x <- rcn(n1,eps,vc); y <- rcn(n2,eps,vc) + Delta
if(wilcox.test(y,x)$p.value <= alpha){indwil <- indwil + 1}
if(t.test(y,x,var.equal=T)$p.value <= alpha){indt <- indt + 1}
}
powwil <- sum(indwil)/nsims; powt <- sum(indt)/nsims
return(c(powwil,powt))}
Notice that power is computed at the sequence of alternatives Δ =− 3 ,− 2 ,...,3.
For our run, here are the empirical powers:
Δ − 3 − 2 − 1 0 1 2 3
MWW test 0.9993 0.9856 0.6859 0.0527 0.6889 0.9874 0.9988
t-test 0.7245 0.4411 0.1575 0.0465 0.1597 0.4318 0.7296
Clearly for this situation the MWW test is much more powerful than thet-test. It
is not surprising since the contaminated normal distribution has heavy tails and the
t-test is impaired by the high percentage of outliers. Further, this agrees with the
ARE between the MWW andt-tests for contaminated normal distributions. The
empirical powers for Δ = 0 are the empirical levels that are close to the nominal
α=0.05. For both tests, the powers increase as Δ moves in either direction from
0, as they should.
EXERCISES
10.4.1.By considering the asymptotic power lemma, Theorem 10.4.2, show that
the equal sample size situationn 1 =n 2 is the most powerful design among designs
withn 1 +n 2 =n,nfixed, when level and alternatives are also fixed.
Hint:Show that this problem is equivalent to maximizing the function
g(n 1 )=n 1 (n−n 1 )
n^2,and then obtain the result.
10.4.2.Consider the asymptotic version of thet-test for the hypotheses (10.4.4)
which is discussed in Example 4.6.2.(a)Using the setup of Theorem 10.4.2, derive the corresponding asymptotic power
lemma for this test.(b)Use your result in part (a) to obtain expression (10.4.25).10.4.3.In the power study presented in Section 10.4.4, the empirical powers at
Δ = 0 are empirical levels. Find 95% confidence intervals for the true levels based
on the empirical levels. Do they contain the nominal levelα=0.05?
10.4.4.In the power study of Section 10.4.4, determine (by simulation) the neces-
sary common sample size so that the Wilcoxon MWW test has approximately 80%
power to detect Δ = 1.