estimation with heterogeneous sequential k-out-of-n system lifetimes

when the component lifetimes are independent and identically distribution, the order statistics are suitable for describing the system lifetime. notice that here failing a component does not change the distribution functions of lifetimes of surviving components. sequential order statistics, is an extension of the usual order statistics and used for modeling lifetimes of sequential r -out-of- n systems, coming from heterogeneous exponential distributions are considered. the problem of hypotheses testing for exponential populations on the basis of multiple independent sequential order statistics samples under a conditionally proportional hazard rate model via a bayesian and classical approach is studied. via the likelihood approach, statistical procedures including estimation, either point or interval of the parameters as well as the generalized likelihood ratio are derived for testing homogeneity of the exponential populations. approximate confidence intervals and fisher information are derived on the basis of observed multiply system lifetimes.