An approach for the Bayesian estimation in the case of ordered piecewise constant failure rates

In this paper we present an approach for the Bayesian estimation of piecewise constant failure rates under the constraint that the constant value of the failure rate in an interval of time is greater than a function of its values in the prior intervals. We apply this approach to the estimation of piecewise constant failure rates for conditional IFR, IFRA and NBU distributions. The prior distribution for the failure rate in each interval is specified through gamma distributions with functions of the failure rate values corresponding to the rest of the intervals as location parameters. Using this approach the prior distribution parameters have interpretations through prior means and variances of the values of the piecewise constant failure rate. The posterior distributions and expected values can be found in terms of gamma functions, without the necessity of numerical integrations. We apply this approach to a model for reliability estimation when two operational modes exists and the number of failures in each operational mode is unknown. Finally a numerical example is presented in which simulations of posterior densities are carried out.