Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

582 Nonparametric and Robust Statistics

whereφ(z) is the pdf of a standard normal random variable. As shown in Section

3.4, the variance ofεiisb^2 (1 + (σ^2 c−1)). Also,τs=(b

√
π/2)/[1− +(/σc)].
Thus the ARE is

ARE(S, t)=

2

π

[(1 + (σ^2 c−1)][1− +(/σc)]^2. (10.2.31)

For example, the following table (see Exercise 6.2.6) shows the AREs for various
values of whenσcissetat3.0:

0 0.01 0.02 0.03 0.05 0.10 0.15 0.25

ARE(S,t) 0.636 0.678 0.718 0.758 0.832 0.998 1.134 1.326

Note: if increases over the range of values in the table, then the contamination
effect becomes larger (generally resulting in a heavier-tailed distribution) and as the
table shows, the sign test becomes more efficient relative to thet-test. Increasing
σchas the same effect. It does take, however, withσc= 3, over 10% contamination
before the sign test becomes more efficient than thet-test.

10.2.2 Estimating Equations Based on the Sign Test

In practice, we often want to estimateθ, the median ofXi, in Model (10.2.1). The

associated point estimate based on the sign test can be described with a simple

geometry, which is analogous to the geometry of the sample mean. As Exercise

10.2.6 shows, the sample meanXis such that

X=Argmin

√

√

√

√

∑n

i=1

(Xi−θ)^2. (10.2.32)

The quantity

√∑n

i=1(Xi−θ)

(^2) is the Euclidean distance between the vector of
observationsX=(X 1 ,X 2 ,...,Xn)′and the vectorθ 1. If we simply interchange
the square root and the summation symbols, we go from the Euclidean distance to
theL 1 distance. Let
̂θ=Argmin
∑n
i=1
|Xi−θ|. (10.2.33)
To determineθ̂, simply differentiate the quantity on the right side with respect to
θ(as in Chapter 6, define the derivative of|x|to be 0 atx= 0). We then obtain
∂
∂θ
∑n
i=1
|Xi−θ|=−
∑n
i=1
sgn(Xi−θ).
Setting this to 0, we obtain the estimating equations (EE)
∑n
i=1
sgn(Xi−θ)=0, (10.2.34)