Abstract
In this paper, we propose and study a new bivariate Weibull model, called Bi-levelWeibullModel, which arises when one failure occurs after the other. Under some specific regularity conditions, the reliability function of the second event can be above the reliability function of the first event, and is always above the reliability function of the transformed first event, which is a univariate Weibull random variable. This model is motivated by a common physical feature that arises fromseveral real applications. The two marginal distributions are a Weibull distribution and a generalized three-parameter Weibull mixture distribution. Some useful properties of the model are derived, and we also present the maximum likelihood estimation method. A real example is provided to illustrate the application of the model.
Highlights
Consider two ordered events A and B with event A occurring before event B, and event B may not occur within a time window right after event A
Motivated by the maintenance tasks in aerospace industry, we propose a new bivariate Weibull mode, i.e., Bi-level Weibull Model, to model the case that the reliability function of the second-level event is greater than that of the first-level event
And the conditional survival function P (T2 > t | T1 = t1) for t > δ + t1: P (T2 > t|T1 = t1) = e . −θ2(t−δ)σ2 +θ2tσ11. We show that this BLW (θ1, θ2, σ1, σ2, δ) can be used to model two ordered events with the survival function of the second event being always above that of the first event
Summary
Consider two ordered events A and B with event A occurring before event B, and event B may not occur within a time window right after event A. Motivated by the real aerospace problems as described below, we propose a new Bi-level Weibull model in this paper. Some recent references on bivariate Weibull distributions include D. Motivated by the maintenance tasks in aerospace industry, we propose a new bivariate Weibull mode, i.e., Bi-level Weibull Model, to model the case that the reliability function of the second-level event is greater than that of the first-level event. The marginal distribution for the first-level event is a two-parameter Weibull distribution, and the marginal distribution for the second-level event is a generalized threeparameter Weibull mixture distribution This model can be used quite effectively if the bivariate data show a non-constant hazard rate. The distributional properties and parameter-estimation methods of the bi-level Weibull model are studied in Section 3 and Section 4, respectively.
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