Estimation with Sequential Order Statistics from Exponential Distributions

The lifetime of an ordinary k-out-of-n system is described by the (n−k+1)-st order statistic from an iid sample. This set-up is based on the assumption that the failure of any component does not affect the remaining ones. Since this is possibly not fulfilled in technical systems, sequential order statistics have been proposed to model a change of the residual lifetime distribution after the breakdown of some component. We investigate such sequential k-out-of-n systems where the corresponding sequential order statistics, which describe the lifetimes of these systems, are based on one- and two-parameter exponential distributions. Given differently structured systems, we focus on three estimation concepts for the distribution parameters. MLEs, UMVUEs and BLUEs of the location and scale parameters are presented. Several properties of these estimators, such as distributions and consistency, are established. Moreover, we illustrate how two sequential k-out-of-n systems based on exponential distributions can be compared by means of the probability P(X < Y). Since other models of ordered random variables, such as ordinary order statistics, record values and progressive type II censored order statistics can be viewed as sequential order statistics, all the results can be applied to these situations as well.