Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

586 Nonparametric and Robust Statistics

10.2.9.LetX 1 ,X 2 ,...,Xnbe a random sample that follows the location model
(10.2.1). In this exercise we want to compare the sign tests andt-test of the hy-
potheses (10.2.2); so we assume the random errorsεiare symmetrically distributed
about 0. Letσ^2 =Var(εi). Hence the mean and the median are the same for this
location model. Assume, also, thatθ 0 = 0. Consider the large sample version of
thet-test, which rejectsH 0 in favor ofH 1 ifX/(σ/

√

n)>zα.

(a)Obtain the power function,γt(θ), of the large sample version of thet-test.

(b)Show thatγt(θ) is nondecreasing inθ.

(c)Show thatγt(θn)→ 1 −Φ(zα−σθ∗), under the sequence of local alternatives

(10.2.13).

(d)Based on part (c), obtain the sample size determination for thet-test to detect

θ∗with approximate powerγ∗.

(e)Derive the ARE(S, t) given in (10.2.27).

10.3Signed-RankWilcoxon..........................

LetX 1 ,X 2 ,...,Xnbe a random sample that follows Model (10.2.1). Inference forθ
based on the sign test is simple and requires few assumptions about the underlying
distribution ofXi. On the other hand, sign procedures have the low efficiency of 0. 64
relative to procedures based on thet-test given an underlying normal distribution.
In this section, we discuss a nonparametric procedure that does attain high efficiency
relative to thet-test. We make the additional assumption that the pdff(x)ofεi
in Model (10.2.1) is symmetric; i.e.,f(x)=f(−x), for allxsuch that−∞<x<
∞. HenceXiis symmetrically distributed aboutθ. Thus, by Theorem 10.1.1, all
location parameters are identical.
First, consider the one-sided hypotheses

H 0 :θ=0versusH 1 : θ> 0. (10.3.1)

There is no loss of generality in assuming that the null hypothesis isH 0 : θ=0,for
if it wereH 0 :θ=θ 0 , we would consider the sampleX 1 −θ 0 ,...,Xn−θ 0. Under
a symmetric pdf, observationsXithat are the same distance from 0 are equilikely
and hence should receive the same weight. A test statistic that does this is the
signed-rank Wilcoxongiven by

T=

∑n

i=1

sgn(Xi)R|Xi|, (10.3.2)

whereR|Xi|denotes the rank ofXiamong|X 1 |,...,|Xn|, where the rankings are
from low to high. Intuitively, under the null hypothesis, we expect half of theXisto
be positive and half to be negative. Further, the ranks are uniformly distributed on
the integers{ 1 , 2 ,...,n}. Hence values ofTaround 0 are indicative ofH 0 .Onthe