Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.9. Robust Concepts 651

(a)Determine the natural estimator of the variance by writing the defining equa-

tion at the empirical cdfFn(t), forX 1 −X,...,Xn−Xiid with cdfFX(t),

and solving forV(Fn).

(b)As in Exercise 10.9.6, write the defining equation for the variance functional

at the contaminated cdfFx,

(t).

(c)Then derive the influence function by implicit differentiation of the defining
equation in part (b).
10.9.8.Show that the inverse of the cdfFx,
(t) given in expression (10.9.17) is
correct.

10.9.9.Let IF(x) be the influence function of the sample median given by (10.9.20).
DetermineE[IF(X)] and Var[IF(X)].

10.9.10. Letx 1 ,x 2 ,...,xn be a realization of a random sample. Consider the
Hodges–Lehmann estimate of location given in expression (10.9.4). Show that the
breakdown point of this estimate is 0.29.
Hint:Suppose we corruptmdata points. We need to determine the value ofmthat
results in corruption of one-half of the Walsh averages. Show that the corruption
ofmdata points leads to

p(m)=m+

(

m

2

)

+m(n−m)

corrupted Walsh averages. Hence the finite sample breakdown point is the “correct”
solution of the quadratic equationp(m)=n(n+1)/4.

10.9.11. For anyn×1 vectorv, define the function‖v‖Wby

‖v‖W=

∑n

i=1

aW(R(vi))vi, (10.9.53)

whereR(vi) denotes the rank ofviamongv 1 ,...,vnand the Wilcoxon scores are
given byaW(i)=φW[i/(n+1)] forφW(u)=

√

12[u−(1/2)]. By using the corre-

spondence between order statistics and ranks, show that

‖v‖W=

∑n

i=1

a(i)v(i), (10.9.54)

wherev(1)≤···≤v(n)are the ordered values ofv 1 ,...,vn. Then, by establishing
the following properties, show that the function (10.9.53) is apseudo-normon
Rn.

(a)‖v‖W≥0and‖v‖W=0ifandonlyifv 1 =v 2 =···=vn.

Hint:First, because the scoresa(i) sum to 0, show that

∑n

i=1

a(i)v(i)=

∑

i<j

a(i)[v(i)−v(j)]+

∑

i>j

a(i)[v(i)−v(j)],

wherejis the largest integer in the set{ 1 , 2 ,...,n}such thata(j)<0.