This exercise involves some of the details in Example 14.8. (i) Show that the functional is

This exercise involves some of the details in Example 14.8.

(i) Show that the functional θ is a special case of (14.48).

(ii) Verify that θ(Fn) = n

−2



n i=1



n j=1 1(Xi+Xj>0), which is a V -

statistic.

(iii) Verify (14.51) and (14.52) and thus conclude that the asymptotic conditional distribution of

√

n(U

∗

n

−Un) given X1, . . . , Xn is the same as the asymptotic distribution of

√

n(Un − θ).

(iv) What is the asymptotic distribution of

√

n(Un − θ)?