Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates

k-out-of-n systems frequently appear in applications. They consist of n components of the same kind with independent and identically distributed life-lengths. The life-length of such a system is described by the (n–k+1)-th order statistic in a sample of size n when assuming that remaining components are not affected by failures. Sequential order statistics are introduced as a more flexible model to describe sequential k-out-of-n systems in which the failure of any component possibly influences the other components such that their underlying failure rate is parametrically adjusted with respect to the number of preceding failures. Useful properties of the maximum likelihood estimators of the model parameters are shown, and several tests are proposed to decide whether the new model is the more appropriate one in a given situation. Moreover, for specific distributions, e.g. Weibull distributions, simultaneous maximum likelihood estimation of the model parameters and distribution parameters is considered.