Asymptotic linearity of linear rank statistic in the case of symmetric heteroscedastic variables

Let Y1, . . . , Yn be independent but not identically distributed random variables with densities f1, . . . , fn symmetric around zero. Suppose c1,n, . . . , cn,n are given constants such that ∑ i ci,n = 0 and ∑ i c 2 i,n = 1. Denote the rank of Yi −∆ci,n for any ∆ ∈ R by R(Yi −∆ci,n) and let an(i) be a score defined via a score function φ. We study the linear rank statistic Sn(∆) = n ∑ i=1 ci,nan[R(Yi −∆ci,n)] and show that Sn(∆) is asymptotically uniformly linear in the parameter ∆ in any interval [−C,C], C > 0.