Dependence properties of order statistics

• DOI: 10.1016/j.jspi.2007.09.013
• Publication Date: 2007-12-04T09:18:41Z
• Authors: Taizhong Hu; Huaihou Chen
• Keywords: 0502 economics and business; 05 social sciences; 0101 mathematics; 01 natural sciences

Let X1:n⩽X2:n⩽⋯⩽Xn:nX1:n⩽X2:n⩽⋯⩽Xn:n denote the order statistics of independent random variables X1,X2,…,XnX1,X2,…,Xn with possibly nonidentical distributions for each n . For fixed 1⩽j1 y}Ai,m,y={Xi:m>y}, then Δi(y)≡P[Xj1:n>x1,Xj2:n>x2,…,Xjr:n>xr|Ai,m,y]Δi(y)≡P[Xj1:n>x1,Xj2:n>x2,…,Xjr:n>xr|Ai,m,y] is increasing in y , and that if Ai,m,yAi,m,y is either {Xi:m>y}{Xi:m>y} or {Xi:m⩽y}{Xi:m⩽y}, then Δi(y)Δi(y) is decreasing in i for fixed y<x1<⋯<xry<x1<⋯<xr. It is also shown that if each XkXk has a continuous distribution function, and if Ai,m,yAi,m,y is either {Xi:m=y}{Xi:m=y} or {Xi-1:m<y<Xi:m}{Xi-1:m<y<Xi:m}, then Δi(y)Δi(y) is decreasing in i for fixed y<x1<⋯<xry<x1<⋯<xr, where Xm+1:m=+∞Xm+1:m=+∞. In particular, we obtain that RTI (Xj:n|Xi:m)(Xj:n|Xi:m) for j-i⩾max{n-m,0}j-i⩾max{n-m,0} and LTD (Xj:n|Xi:m)(Xj:n|Xi:m) for j-i⩽min{n-m,0}j-i⩽min{n-m,0}. We thus extend the main results in Boland et al. [1996. Bivariate dependence properties of order statistics. J. Multivariate Anal. 56, 75–89] and in Hu and Xie [2006. Negative dependence in the balls and bins experiment with applications to order statistics. J. Multivariate Anal. 97, 1342–1354].