Chapter 7 Multivariate Ageing Concepts

The notions of ageing and classification of life distributions based on them are of importance in stochastic modelling and reliability analysis for components and systems. In this chapter, we make a systematic study of various ageing concepts relating to multi-component systems where lifetime is thought of as a discrete random variable. To begin with, some stochastic orders for multivariate discrete random variables that are of use in the sequel are introduced. This is followed by a discussion of various versions of no-ageing property in the multivariate framework. The distributions characterized by the no-ageing property are then derived. The multivariate increasing (decreasing) hazard rate notion is presented using the survival function, scalar hazard rate, vector hazard rate and conditional hazard rate. Bayesian approach to defining multivariate ageing classes is substantially different from the conventional approach. Different versions of increasing hazard rate and decreasing mean residual life are reviewed. Increasing (decreasing) hazard rate average and new better than used in hazard rate classes and their properties are discussed next. Various definitions of decreasing (increasing) multivariate mean residual life notions are described with examples. Different properties of these classes are discussed. Generalizations of the NBU, NBUE and HNBUE classes under different assumptions for reckoning them and the mutual implications between them are discussed subsequently. Covariance and product moment of residual lives form important elements in the analysis of ageing and dependence in multi-component devices. The classes of life distributions based on the monotonicity of the product moment of residual life are discussed at the end.