Estimating the parameters of parametric lifetime distributions through an efficient acceptance–rejection sampler

The three-parameter (3-p) Weibull distribution is an extremely important distribution to characterise the statistical behaviour of a large number of real world phenomenons. It is also useful as a failure model in analysing the reliability of different types of mechanical and electrical components/systems. Successful applications of the distribution rely on an accurate estimation of its three parameters because it directly affects the reliability and lifetime analysis. Due to the intricate system of nonlinear equations and the complexity of the likelihood function, derivative-based optimisation methods may fail to converge. Thus, an efficient and effective method for estimating the parameters of the model is important from the practical viewpoint. In this paper, an optimisation scheme based on an acceptance–rejection (AR) mechanism coupled with an elegant nested sampling (NS) technique is proposed to tackle this problem. The idea is to gradually approach the region of optimal solutions through an efficient sampling technique and a reweighting scheme. The AR-NS algorithm allows a good exploration of the parameter space and converges towards higher likelihood regions by decreasing progressively a pre-specified tolerance threshold. The proposed approach gives the entire distributions of the optimal estimates rather than a single point estimates. To demonstrate the practicality and the efficiency of the proposed approach, numerous numerical examples using simulated data and real-world engineering cases will be given. The obtained results show that the AR-NS algorithm is a suitable method for estimating the parameters of lifetime distributions using different distances.