MULTIVARIATE STOCHASTIC COMPARISONS OF SEQUENTIAL ORDER STATISTICS

In this article we investigate conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain stochastic comparisons of sequential order statistics in the multivariate likelihood ratio, the multivariate hazard rate, and the usual multivariate stochastic orders. Some applications of the main results are also given.

Preservation of multivariate stochastic orders under multivariate Poisson shock models

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i (t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γ m are independent of Mk = (Mk, 1, · ··, Mk,m ), the number of shocks each component in system k can sustain without failure. Let Zk,i be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm such that some stochastic orders between M 1 and M 2 are transformed into some stochastic orders between Z 1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γ m are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.

Preservation of multivariate stochastic orders under multivariate Poisson shock models

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i(t), t > 0}, i = 1, ···, m. Assume that Γ1, · ··, Γm are independent of Mk = (Mk,1, · ··, Mk,m), the number of shocks each component in system k can sustain without failure. Let Zk,i be the lifetime of component i in system k. We find conditions on processes Γ1, · ··, Tm such that some stochastic orders between M1 and M2 are transformed into some stochastic orders between Z1 and Z2. Most results are obtained under the assumption that Γ1, · ··, Γm are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.