Order Statistics And ��ά��Ŭ�� Random Vector Dependence Structure

In everyday life, people usually use random variables or random vectors to describe uncertain factors about things and their development. While the dependcncc and restrictions among the uncertain factors, the random variables or random vcctors, that influence things and their development become the main research objects of the stochastic dependence theory. The purpose of this paper is to investigate several positive and/or negative dependence properties about two special multivariate random vectors, ordinary order statistics and the Bernoulli random vectors.Firstly, for occupancy number vector B = (B₁, B-2, ..., B_n) in the balls and bins experiment, we prove that IE[φ(B_L)|B_I ≥ b_I,B_J≤ b_J,B_K = b_K] is decreasing in nonnegative vectors b_I,b_J and b_K for all increasing functions φ : R^|L| →R, where the conditional expectations arc assumed to exist, I, .J, K, L are four disjoint subsets of {1,2,... , n}, and one or two of 7, .7 and K may be an empty set. In particular, We get that B is NRD, NRTD and NLTD. By applying such properties, we investigate further dependence structures of order statistics X1:n ≤ X₂:n <...< Xn:n of n independent random variables X₁, X-2,... ,X_n with possibly different distributions. For 1 ≤ i < j₁ < j₂ < ... < j_r < n and fixed (x₁,... ,x_r), we show that IP (X_j1:n> X₁,X_j2:n > x₂,....,X_jr:n > x_r|X_i:n > s) is increasing in 5, and that if event A_i,s is either {X_i:n> s} or {X_i:n ≤ s} then IP(X_j1:n > x₁,X_j2:n > x₂,... ,X_jr:n > x_r|A_i,s) is decreasing in i for fixed s. It is also shown that in this situation, if each X_k has a continuous distribution function then A_i,s can be chosen to be either {X_i1:n < s < X_i:n} or {X_i:n = s}. We thus complement and extend some results in Dubhashi and Ranjan (1998) and Boland ct al. (1996).Secondly, multivariate Bernoulli random vectors have many applications in several fields. Their probability mass functions are a special weighted tree —probability tree. We utilize a newly developed tool, majorization with respect to weighted trees, to establish the sufficient and necessary conditions for multivariate Bernoulli random vectors to possess some negative and positive dependence properties such as POD, NOD, SPOD, SNOD. Besides, we get the...