Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.2. Sample Median and the Sign Test 581

Note that this is the same relative efficiency that was discussed in Example 6.2.5
when the sample median was compared to the sample mean. In the next two
examples we revisit this discussion by examining the AREs whenXihas a normal
distribution and then a Laplace (double exponential) distribution.

Example 10.2.2(ARE(S, t): normal distribution).SupposeX 1 ,X 2 ,...,Xnfollow
the location model (10.1.4), wheref(x)isaN(0,σ^2 ) pdf. ThenτS=(2f(0))−^1 =
σ

√

π/2. Hence the ARE(S, t)isgivenby

ARE(S, t)=

σ^2

τS^2

=

σ^2

(π/2)σ^2

=

2

π

≈ 0. 637. (10.2.28)

Hence at the normal distribution the sign test is only 64% as efficient as thet-test.
In terms of sample size at the normal distribution, thet-test requires a smaller
sample, 0. 64 ns,wherensis the sample size of the sign test, to achieve the same
power as the sign test. A cautionary note is needed here because this is asymptotic
efficiency. There have been ample empirical (simulation) studies that give credence
to these numbers.
Example 10.2.3(ARE(S, t) at the Laplace distribution).For this example, con-
sider Model (10.1.4), wheref(x) is the Laplace pdff(x)=(2b)−^1 exp{−|x|/b}for
−∞<x<∞andb>0. ThenτS=(2f(0))−^1 =b, whileσ^2 =E(X^2 )=2b^2.
Hence the ARE(S, t)isgivenby

ARE(S, t)=

σ^2

τS^2

=

2 b^2

b^2

=2. (10.2.29)

So, at the Laplace distribution, the sign test is (asymptotically) twice as efficient
as thet-test. In terms of sample size at the Laplace distribution, thet-test requires
twice as large a sample as the sign test to achieve the same asymptotic power as
the sign test.
Recall from Example 6.3.4 that the sign test is the scores type likelihood ratio
test when the true distribution is the Laplace.
The normal distribution has much lighter tails than the Laplace distribution,
because the two pdfs are proportional to exp{−t^2 / 2 σ^2 }and exp{−|t|/b}, respec-
tively. Based on the last two examples, it seems that thet-test is more efficient
for light-tailed distributions while the sign test is more efficient for heavier-tailed
distributions. This is true in general and we illustrate this in the next example
where we can easily vary the tail weight from light to heavy.

Example 10.2.4(ARE(S, t) at a family of contaminated normals).Consider the
location Model (10.1.4), where the cdf ofεiis the contaminated normal cdf given
in expression (3.4.19). Assume thatθ 0 = 0. Recall that for this distribution, (1− )
proportion of the time the sample is drawn from aN(0,b^2 ) distribution, while
proportion of the time the sample is drawn from aN(0,b^2 σ^2 c) distribution. The
corresponding pdf is given by

f(x)=

1 −

b

φ

(x

b

)

+

bσc

φ

(

x

bσc

)

, (10.2.30)