Negative dependence in the balls and bins experiment with applications to order statistics

• Authors: Taizhong Hu;Chaode Xie
• Venue: Journal of Multivariate Analysis
• Year: 2006

Dependence properties of occupancy numbers in the balls and bins experiment are studied. Applying such properties, we investigate further dependence structures of order statistics X1:n ≤ X2:n ≤...≤ Xn:n of n independent random variables X1, X2,...,Xn with possibly different distributions. For 1 ≤ i j1 j2 jr ≤ n and fixed (x1,...,xr), we show that P(Xj1:n x1, Xj2:n x2,...,Xjr:n Xr|Xi:n s) is increasing in s, and that if event Ai,s is either {Xi:n s} or {Xi:n ≤ s} then P (Xj1:n x1, Xj2:n x2,...,Xjr:n xr|Ai,s) is decreasing in i for fixed s. It is also shown that in this situation, if each random variable Xk has a continuous distribution function and if Ai,s is either {Xi-1:n s Xi:n} or {Xi:n = s} then P(Xj1:n x1, Xj2:n x2,...,Xjr:n xr|Ai,s) is decreasing in i for fixed s. We thus complement and extend some results in Dubhashi and Ranjan [Balls and bins: a study in negative dependence, Random Struct. Algorithms 13 (1998) 99-124] and Boland et al. [Bivariate dependence properties and order statistics, J. Multivar. Anal. 56 (1996) 75-89].