Mohtashami Borzadaran, G. R. , Ahmadi Mosayeb and Charkhi Ali
The 9th Seminar on Probability and Stochastic Processes, University of Sistan and Baluchestan
Publication year: 2013


Ageing concepts of the life times of a device or system is an important problem which describes how a device ages with times. These have direct connections with the behavior of reliability measures such as failure rate and mean residual life functions. They have an important role in areas like maintenance planning, replacement policy and many other reliability related decisions. Based on cases components of systems are mutual independent or dependent, identical distributions or non-identical distributed in univariate and multivariate cases. Many testing hypothesis developed for test exponentiality ageing against di erent ageing alternative. Among various view, we are interested in stochastic ageing and ageing classes of univariate distributions. We have focused on classes of distributions based on notions of ageing IFR the best known and familiar but we shall meet several other ageing classes and study their inter-relationships. Closure properties of ageing classes and characterization aspects are discussed here. Sometimes, for modeling statistical data, using standard distributions was not found appropriate. Due to this idea, Fisher (1934) de ned weighted distributions. Later Rao (1965) used weighted distribution in some modeling problem. After them, lots of researches published and extended in a wide range the weighted versions of distributions. Order statistics, record values, k-record, truncated distributions, proportional hazard, reversed proportional hazard, probability weighted moments, sized biased, Jones model and huge of weights are applied by many authors as examples of weighted distributions. A famous general case is weights that are functions of the distribution functions. They subsume lots of familiar cases also. Finding structural relationships between the original and weighted random variables for class of life time distributions and ageing classes is one the main theme of the content of this paper. The nal section of the paper organized by various stochastic order that is the order on the set of probability models. Concentration on most orders that are helpful is working with the ageing classes are obtained as our focus. Extending the results of this section to weighted version (in a general case) are achieved and some cases which more popular are presented