Analysis of incomplete data in competing risks model : Masami Miyakawa. IEEE Trans. Reliab.R-33 (4) 293 (October 1984)

Abstract

Ageing concepts that distinguish whether a device is ageing positively or negatively form an important component of reliability analysis. We begin this chapter by presenting the importance and use of ageing characteristics. Stochastic orders provide means of comparing the ageing patterns and also help in defining certain notions in this context. The usual stochastic order, hazard rate order, likelihood ratio order, mean residual life order, increasing convex and concave orders, variance residual life order, and some others are all defined for discrete random variables and their inter-relationships are given. Distributions can be distinguished by means of no memory, positive memory and negative memory. We introduce measures of memory and discuss some stochastic orders based on them. The ageing concepts can be classified as those based on hazard rates, characteristics of residual life, and the survival function. Among those in terms of the hazard rate, increasing hazard rate, increasing hazard rate (2), increasing hazard rate average of two different types, new better than used in hazard rate, new better than used in hazard rate average and their duals are discussed. Based on residual life, the decreasing mean residual life, decreasing mean residual life in harmonic average, used better than aged, used better than aged in expectation, decreasing variance residual life classes and their duals are discussed. Following these, we discuss the new better than used (NBU), NBU-1, NBU-2, new better than used in convex order, new better than used in probability generating function order, new better than used in expectation, harmonically new better than used in expectation (HNBUE), HNBUE-k and HNBUE (3) classes. In a similar manner various concepts based on reliability functions in reversed time also form part of this chapter. In the case of all ageing notions, their properties are studied in detail. Considering the role of weighted distributions and equilibrium models in particular, we have established the inter-relationships between the ageing classes of the baseline and weighted models. The final section is on various methods for measuring the relative ageing of two devices. Stochastic orders among ageing notions, specific ageing factor, relative ageing factor and ageing intensity function and their properties are also discussed in this connection.