We study some properties for multivariate weighted distributions related to reliability measures, ordering, characterization and dependence properties, by compiling and extending previous results given by different authors. We pay special attention to the multivariate size biased and equilibrium distributions, and propose a new definition for the multivariate equilibrium distribution.

Multivariate weighted distributions: a review and some extensions

Classical and Contagious Discrete Distributions

Cumulative probabilities of a Poisson distribution can be written in terms of incomplete gamma function where the parameter of the gamma function is an integer. From this definition a new generalization of the Poisson distribution is obtained with two parameters. Asymptotic behavior of this distribution is shown to be normal. Some order properties of this distribution are also studied.

A Generalized Poisson Distribution

In the present work we study the asymptotic distribution of the age and residual life in a renewal process. The bivariate distribution so derived is Schur-constant with marginal distributions as equilibrium discuss the reliability properties and the copula. The bivariate ageing properties and some stochastic orders connecting them are explored. It is shown that various time dependent measures of association and dependence concepts can be inferred from the ageing properties of the baseline distribution.

Modelling lifetimes with bivariate Schur-constant equilibrium distributions from renewal theory

We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CD8AC5C74F3780E8EDBB8B62275225C9/S0001867820000063a.pdf/div-class-title-failure-rate-properties-of-parallel-systems-div.pdf

S.P. Mukherjee

Papers

Ch. 4. Equilibrium distribution — its role in reliability theory