Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.3. Signed-Rank Wilcoxon 595

we then have that

1 −α = Pθ[cW 1 <T+(θ)<n−cW 1 ]

= Pθ[WcW 1 +1≤θ<Wm−cW 1 ], (10.3.33)

wherem=n(n+1)/2 denotes the number of Walsh averages. Therefore, the interval
[WcW 1 +1,Wm−cW 1 )isa(1−α)100% confidence interval forθ.
We can use the asymptotic null distribution ofT+, (10.3.10), to obtain the
following approximation tocW 1. As shown in Exercise 10.3.6,

cW 1 ≈

n(n+1)

4

−zα/ 2

√

n(n+ 1)(2n+1)

24

−

1

2

, (10.3.34)

where Φ(−zα/ 2 )=α/2. In practice, we use the closest integer tocW 1.
In R, this confidence interval is computed by the R functionwilcox.test.For
instance, for the Darwin data let the R vectordscontain the paired differences,
Cross−Self. Then the callwilcox.test(ds,conf.int=T,conf.level=.95)com-
putes a 95% confidence interval for the median of the differences. Its computation
results in the interval (0. 5000 , 5 .2125). Hence, with confidence 95%, we estimate
that cross-fertilizedzea maysare between 0.5 to 5.2 inches taller than self-fertilized
ones.

10.3.4 MonteCarloInvestigation....................

The AREs derived in this chapter are asymptotic. In this section, we describe
Monte Carlo techniques which investigate the relative efficency between estimators
for finite sample sizes. Comparisons are performed over families of distributions
and a selection of sample sizes. Each combination of distribution and sample size
is referred to as asituation. We also select a simulation sizens, which is usually
quite large. We next describe a typical simulation to investigate the relative efficency
between two estimators.
For notation, letX 1 ,...,Xnbe a random sample that follows the location model,
(10.2.1), i.e.,
Xi=θ+ei,i=1,...,n, (10.3.35)

whereei’s are iid with pdff(x)andf(x) is symmetric about 0. For our discussion,

consider the case of two location estimators ofθ, which we denote byθ̂ 1 and̂θ 2.
Since these are location estimators, we further assume without loss of generality
that the trueθ=0.
Letndenote the sample size and letf(x) denote the pdf for a given situation.
Thennsindependent random samples of sizenare generated fromf(x). For the
ith sample, denote the estimates bŷθ 1 iand̂θ 2 i,i=1,...,ns. For the estimator̂θj,
consider the mean square error over the simulations given by

MSEj=

1

ns

∑ns

i=1

θ̂^2 ji,j=1, 2. (10.3.36)