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Stochastic Comparison of Some Wear Processes

Published online by Cambridge University Press:  27 July 2009

Franco Pellerey  and
Moshe Shaked
Franco Pellerey
Affiliation:
Dipartimento di Matematica, Universitá di Milano, via L. Cicognara, 7, 20129 Milano, Italy
Moshe Shaked
Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721
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Abstract

Let [NA(t), t≥0] and [NB(t), t≥0] be two counting processes with independent interarrivals Ai and Bi, respectively, i ∈ + ≡ [1,2,… ], and let [Xi, i ∈ +] and [Yi, i ∈ +] be two sequences of independent non-negative random variables. Consider the two compound processes [U(t), t ≥ 0] and [V(t), t ≥ 0] defined as U(t) = Xi, t ≥ 0, and V(t) = Yi, t ≥ 0. Then it is well known (and not hard to verify) that if the Ai's are independent of the Xi's, and if the Bi's are independent of the yi's, and if Ai ≥st Bi and Xi ≤st Yi, for all i ∈ +, then [U(t), t ≥ 0] ≤st [V(t), t ≥ 0]. In this paper we provide conditions under which this stochastic order relation holds even if Ai ≤st Bi and/or Xi ≥st Yi for all i ∈ +. In fact, we derive the stochastic order relationships among processes that are much more general than the cumulative ones described above. For example, it is not necessarily assumed that the Ai's are independent of the Xi's or that the Bi's are independent of the Yi's. Examples that illustrate the applications of the theory are included.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 3 , July 1993 , pp. 421 - 435
Copyright
Copyright © Cambridge University Press 1993

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