Estimation in reliability models with distribution having bathtub shaped hazard function

Despite many reliability models with distribution having bathtub shaped hazard function have been proposed in the literature, the estimation of the unknown parameters of such reliability models is one of the most important problems faced by applied statisticians. Modified Weibull distribution with bathtub shaped hazard function is a flexible distribution. Estimating the parameters of this distribution is quite problematic. In general, the most frequently used method in the literature to estimate the parameters of reliability models with distribution having bathtub shaped hazard rate function is Maximum Likelihood (ML) method. For bathtub hazard rate distributions, the ML equations involve nonlinear functions of parameters and will not yield closed form solution. Thus, numerical and iterative methods are needed. Using iterative methods can be problematic for reasons of multiple roots, and nonconvergence of iterations or convergence to wrong values. In this study, 3-Step Modified Maximum Likelihood (MML) and Average methods have been proposed to estimate the parameters of the Modified Weibull distribution which is a flexible distribution with bathtub shaped hazard rate function. Monte Carlo simulation has been performed to compare the efficiency of the proposed methods and the Least Squares (LS) method. Moreover, asymptotic variances for the estimators of the parameters are also given. Simulation results show that the proposed methods are more efficient than the LS method in most of the situations. In general, Average method is better than 3-Step MML and it can be used in general usage. Finally, two applications based on real life data have been given. Keywords: Bathtub hazard function, parameter estimation, Modified Maximum Likelihood (MML), Least Squares (LS), Modified Weibull distribution.