10.6.∗Adaptive Procedures 619
(a)Show that under symmetry the optimal two-sample score function (10.5.26)
satisfies
φf(1−u)=−φf(u), 0 <u<1; (10.5.40)
that is,φf(u) is an odd function about^12. Show that a function satisfying
(10.5.40) is 0 atu=^12.(b)For a two-sample score functionφ(u) that is odd about^12 , define the function
φ+(u)=φ[(u+1)/2], i.e., the top half ofφ(u). Note that the domain ofφ+(u)
is the interval (0,1). Show thatφ+(u)≥0, providedφ(u) is nondecreasing.(c)Assume for the remainder of the problem thatφ+(u) is nonnegative and non-
decreasing on the interval (0,1). Define the scoresa+(i)=φ+[i/(n+1)],
i=1, 2 ,...,n, and the corresponding statisticWφ+=∑ni=1sgn(Xi)a+(R|Xi|). (10.5.41)Show thatWφ+reduces to a linear function of the signed-rank test statistic
(10.3.2) ifφ(u)=2u−1.(d)Show thatWφ+reduces to a linear function of the sign test statistic (10.2.3)
ifφ(u)=sgn(2u−1).
Note: Suppose Model (10.5.39) is true and we takeφ(u)=φf(u), where
φf(u) is given by (10.5.26). If we chooseφ+(u)=φ[(u+1)/2] to generate the
signed-rank scores, then it can be shown that the corresponding test statistic
Wφ+is optimal, among all signed-rank tests.(e)Consider the hypothesesH 0 : θ=0versusH 1 :θ> 0.OurdecisionruleforthestatisticWφ+is to rejectH 0 in favor ofH 1 ifWφ+≥
k,forsomek.WriteWφ+in terms of the anti-ranks, (10.3.5). Show thatWφ+
is distribution-free underH 0.(f)Determine the mean and variance ofWφ+underH 0.(g)Assuming that, when properly standardized, the null distribution is asymp-
totically normal, determine the asymptotic test.10.6∗AdaptiveProcedures
In the last section, we presented fully efficient rank-based procedures for testing and
estimation. As with mle methods, though, the underlying form of the distribution
must be known in order to select the optimal rank score function. In practice,
often the underlying distribution is not known. In this case, we could select a score
function, such as the Wilcoxon, which is fairly efficient for moderate- to heavy-tailed