Estimation in a model of sequential order statistics with ordered hazard rates

Abstract

As a generalization of order statistics from independent and identically distributed random variables, sequential order statistics (SOSs) may be applied as a model for ordered data, when assuming changes of underlying distributions immediately after the occurrences of ordered observations. For example, in the case of a model for k-out-of-n-systems, where the n − k + 1 failures of components in a system of n components occur successively, a change of the respective underlying distribution after a failure is motivated by an increased load put on the remaining components. The corresponding cumulative distribution functions are assumed to have possibly different ordered hazard rates, which are further multiplied by factors, in order to build the hazard rates of the SOSs. These factors are the parameters of interest. Estimation of the parameters is considered by means of maximum likelihood under order restriction, by means of link functions, and in a Bayesian set-up with an order statistics prior.