A random vector X=(X1, X2,…,Xm) is said to be negatively superadditive dependent (NSD) if for every superadditive function Φ, EΦ(X1, X2,…,Xm) ≤EΦ(Y1, Y2,…,Ym), where Y1, Y2,…, Ym are independent with Yi d= Xi for each i. Some basic properties and three structural theorems of NSD are derived and applied to show that a number of well-known multivariate distributions possess the NSD property. Applications are also presented.

The closure property of the up shifted likelihood ratio order under convolutions was first proved by Shanthikumar and Yao (1986) by establishing a stochastic monotonicity property of birth-death processes. Lillo, Nanda and Shaked (2000) made a slight extension of this closure property for any random variables with interval supports by using the result of Shanthikumar and Yao (1986). A new analytic proof of the closure property is given, and the method is applied to establish another result involving the up shifted hazard rate and reversed hazard rate orders.

Results and conditions which quantify the decrease in dependence with lag for a stationary Markov process and enable one to compare the dependence for two stationary Markov processes are obtained. The notions of dependence used in this paper are the supermodular ordering and the concordance ordering. Both discretetime and continuous-time Markov processes are considered. Some applications of the main results are given. In queueing theory, the monotonicity results of the waiting time of the n-th customer as well as the stationary waiting time in an MR/GI/1 queue, and the stationary workload in a Markov-modulated queue are established, thus strengthening previous results while simplifying their derivation. This paper is a continuation of Fang, Hu and Joe [7] and Hu and Joe [10].

Let X1:n ≤ X2:n ≤…≤ Xn:n denote the order statistics of a set of independent and not necessarily identically distributed random variables X1,…, Xn. Under mild assumptions, it is shown that Xk-1:n-1≤lr Xk:n for k=2,…, n if X1≤lr X2≤lr…≤lr Xn, and that Xk:n≤lr Xk:n-1 for k=1,… n-1 if X1≥lr X2≥lr…≥lr Xn, where ≤lr denotes the likelihood ratio order. Concerning the mean residual life order (≤mrl), it is shown that Xn-1:n-1 ≤mrl Xn:n if Xj≤mrl Xn for j=1,…, n-1. Two counterexamples are also given to illustrate that Xk-1:n-1 ≤mrl Xk:n in this case is, in general, not true for k<n.

Dispersion-type orders are introduced and studied. The new orders can be used to compare the variability of the underlying random variables, among which are the usual dispersive order and the right spread order. Connections among the new orders and other common stochastic orders are examined and investigated. Some closure properties of the new orders under the operation of order statistics, transformations and mixtures are derived. Finally, several applications of the new orders are given