618 Nonparametric and Robust Statistics(b)Show that part (a) implies that the efficacy, (10.5.20), is invariant to the
location and varies indirectly with scale.(c)SupposeZis a scale and location transformation of a random variableX; i.e.,
Z=a(X−b), wherea>0and−∞<b<∞. Show thatI(fZ)=a−^2 I(fX).
10.5.10.Consider the scale parameterτφ, (10.5.24), when normal scores are used;
i.e.,φ(u)=Φ−^1 (u). Suppose we are sampling from aN(μ, σ^2 ) distribution. Show
thatτφ=σ.
10.5.11.In the context of Example 10.5.2, obtain the results in expression (10.5.32).
10.5.12. Let the scoresa(i) be generated byaφ(i)=φ[i/(n+1)], fori=1,...,n,
where
∫ 1
0 φ(u)du=0and∫ 1
0 φ(^2) (u)du= 1. Using Riemann sums, with subintervals
of equal length, of the integrals
∫ 1
0 φ(u)duand
∫ 1
0 φ
(^2) (u)du, show that∑n
i=1a(i)≈^0
and
∑n
i=1a
(^2) (i)≈n.
10.5.13. Consider the sign scores test procedure discussed in Example 10.5.4.
(a)Show thatWS=2WS∗−n 2 ,whereWS∗=#j
{
R(Yj)>n+1 2
}
. HenceWS∗is
an equivalent test statistic. Find the null mean and variance ofWS.
(b)Show thatWS∗=#j{Yj>θ∗},whereθ∗is the combined sample median.(c)Supposenis even. LettingWXS∗ =#i{Xi>θ∗}, show that we can tableWS∗
in the following 2×2 contingency table with all margins fixed:Y X
No. items>θ∗ WS∗ WXS∗ n 2
No. items<θ∗ n 2 −WS∗ n 1 −WXS∗ n 2
n 2 n 1 nShow that the usualχ^2 goodness-of-fit is the same asZ^2 S,whereZSis the
standardizedz-test based onWS. This is often calledMood’s median test;
see Example 10.5.4.10.5.14. Recall the data discussed in Example 10.5.3.
(a)Obtain the contingency table described in Exercise 10.5.13.(b)Obtain theχ^2 goodness-of-fit test statistic associated with the table and use
it to test at level 0.05 the hypothesesH 0 :Δ=0versusH 1 :Δ =0.(c)Obtain the point estimate of Δ given in expression (10.5.37).10.5.15.Optimal signed-rank based methods also exist for the one-sample problem.
In this exercise, we briefly discuss these methods. LetX 1 ,X 2 ,...,Xnfollow the
location model
Xi=θ+ei, (10.5.39)
wheree 1 ,e 2 ,...,enare iid with pdff(x), which is symmetric about 0; i.e.,f(−x)=
f(x).