Weighted Rank Regression For Clustered Data Analysis You-Gan Wang and Yudong Zhao Supplementary Materials Appendix A: Asymptotic normality of U W (β ) We will apply the H´ajek projection method to obtain the asymptotic normality of the estimating function U W (β ). The projection of U W (β ) is defined as U ∗ = ∑ N i=1 E(U W | ǫ i ). Note that M = ∑ N i=1 n i = O(N ) because n i is bounded. We first rewrite U W (β ) as U W (β ) = 4 M 2 N  i=1 N  j =i w i w j  k,l x ik I (ǫ ik ≥ ǫ jl ) − 2 M 2 N  i=1 N  j =i w i w j  k,l x ik . Note that E{I (ǫ ik ≥ ǫ jl )| ǫ i } = F jl (ǫ ik ) and E{I (ǫ pq ≥ ǫ jl )|ǫ i } =1/2. From it, we find, after some algebra, E{U W (β )| ǫ i } = 2 M 2  j =i w i w j  k,l (x ik − x jl ){2F jl (ǫ ik ) − 1}. Hence, the projection U ∗ of U W (β ) is U ∗ = 2 M 2 N  i=1  j =i w i w j  k,l (x ik − x jl ){2F jl (ǫ ik ) − 1}. Since E  ∑ j =i w i w j ∑ kl (x ik − x jl )F jl (ǫ ik )  = 1 2 ∑ j =i w i w j ∑ kl (x ik − x jl ), the variance of U ∗ can be easily obtained as Var(U ∗ ) = 16 M 4 N  i=1 Var   j =i w i w j  k,l (x ik − x jl )F jl (ǫ ik )  = 16 M 4 N  i=1 E(ξ i ξ T i ) where ξ i = ∑ j =i w i w j ∑ k,l (x ik − x jl ) {F jl (ǫ ik ) − 0.5}. 1