BIVARIATE MARSHALL–OLKIN EXPONENTIAL SHOCK MODEL

The well-known Marshall–Olkin model is known for its extension of exponential distribution preserving lack of memory property. Based on shock models, a new generalization of the bivariate Marshall–Olkin exponential distribution is given. The proposed model allows wider range tail dependence which is appealing in modeling risky events. Moreover, a stochastic comparison according to this shock model and also some properties, such as association measures, tail dependence and Kendall distribution, are presented. The new shock model is analytically quite tractable, and it can be used quite effectively, to analyze discrete–continuous data. This has been shown on real data. Finally, we propose the multivariate extension of the Marshall–Olkin model that has some intersection with the well-known multivariate Archimax copulas.

The BALM Copula

The class of probability distributions possessing the almost-lack-of-memory property appeared about 20 years ago. It reasonably took place in research and modeling, due to its suitability to represent uncertainty in periodic random environment. Multivariate version of the almost-lack-of-memory property is less known, but it is not less interesting. In this paper we give the copula of the bivariate almost-lack-of-memory (BALM) distributions and discuss some of its properties and applications. An example shows how the Marshal-Olkin distribution can be turned into BALM and what is its copula.

An Extension of the Lack of Memory Property with Characterization Results

Summary In the reliability analysis the lack of memory property plays a pivotal role in conceptualizing some life distribution classes and in unique determination of the exponential distribution. On the other hand quite a few results like constancy of the coefficient of variation of the residual life, linearity of the mean residual life characterize the exponential distribution along with the Lomax distribution. Question that arises is - can there be an extended version of the lack of memory property to tie together the exponential and the Lomax distributions through characterization. The present paper presents an affirmative claim and extends the lack of memory property based on standardization technique. It also presents a stochastic version of this extended property with unique determination of the same life distributions. Attempts have also been made to define this extended lack of memory property in the bivariate set up and indicate the bivariate distributions that satisfy the same.