Generalized Loss of Memory Property and a Multivariate Extension

Abstract

In many practical situations, we observe that the residual life gets independent of age after some shift. For example, at the of operation the life time of a component cannot be described sufficiently well by an exponential distribution as many internal sources of failure are existing. However, once the component has reached, let us say, the full operative status after some time, the component enters a stable state without any essential internal failure modes. During this state the assumption of exponential distribution is justified. We extend the multivariate loss of memory property of Marshall-Olkin (Journal of the American Statistical Association 62: 30–44, 1967) by admitting some shift in the lifetimes of the components. Furthermore, we present four examples of bivariate exponential models that satisfy the here considered generalized loss of memory property.