10.5.∗General Rank Scores 60710.4.5.For the power study of Section 10.4.4, modify the R functionwil2powsim.R
to obtain the empirical powers for theN(0,1) distribution.10.4.6.Use the Central Limit Theorem to show that expression (10.4.30) is true.
10.4.7.For the cutoff indexcof the confidence interval (10.4.31) for Δ, derive the
approximation given in expression (10.4.32).
10.4.8.LetXbe a continuous random variable with cdfF(x). SupposeY=X+Δ,
where Δ>0. Show thatYis stochastically larger thanX.
10.4.9.Consider the data given in Example 10.4.1.
(a)Obtain comparison boxplots of the data.(b)Show that the difference in sample means is 3.11, which is much larger than
the MWW estimate of shift. What accounts for this discrepancy?(c)Show that the 95% confidence interval for Δ usingtis given by (− 2. 7 , 8 .92).
Why is this interval so much larger than the corresponding MWW interval?(d)Show that the value of thet-test statistic, discussed in Example 4.6.2, for
this data set is 1.12 withp-value 0.28. Although, as with the MWW results,
thisp-value would be considered insignificant, it seems lower than warranted
[consider, for example, the comparison boxplots of part (a)]. Why?10.5∗GeneralRankScores
Suppose we are interested in estimating the center of a symmetric distribution
using an estimator that corresponds to a distribution-free procedure. By the last
two sections our choice is either the sign test or the signed-rank Wilcoxon test. If
the sample is drawn from a normal distribution, then of the two we would choose
the signed-rank Wilcoxon because it is much more efficient than the sign test at
the normal distribution. But the Wilcoxon is not fully efficient. This raises the
question: Is there is a distribution-free procedure that is fully efficient at the normal
distribution, i.e., has efficiency of 100% relative to thet-testatthenormal? More
generally, suppose we specify a distribution. Is there a distribution-free procedure
that has 100% efficiency relative to the mle at that distribution? In general, the
answer to both of these questions is yes. In this section, we explore these questions
for the two-sample location problem since this problem generalizes immediately to
the regression problem of Section 10.7. A similar theory can be developed for the
one-sample problem; see Chapter 1 of Hettmansperger and McKean (2011).
As in the last section, letX 1 ,X 2 ,...,Xn 1 be a random sample from the contin-
uous distribution with cdf and pdfF(x)andf(x), respectively. LetY 1 ,Y 2 ,...,Yn 2
be a random sample from the continuous distribution with cdf and pdf, respectively,
F(x−Δ) andf(x−Δ), where Δ is the shift in location. Letn=n 1 +n 2 denote
the combined sample sizes. Consider the hypotheses
H 0 :Δ=0versusH 1 :Δ> 0. (10.5.1)