Parameter Estimation of the Three-Parameter Weibull Distribution Based on an Iterative CDF Method

Parameter estimation of the three-parameter Weibull distribution is an important problem in reliability analysis and statistical modeling. Random right-censored data are widely encountered in engineering practice. Conventional least squares (LS) methods usually construct the empirical cumulative distribution function (CDF) based on rank statistics. However, this empirical assumption cannot adequately capture the nonlinear variation in failure probability with time in the Weibull distribution. To address this limitation, an iterative conditional probability based on conditional failure probability (ICP-CDF) is proposed. The method uses the parameter estimates obtained from the conventional LS approach as initial values, adjusts the ranks of failure data according to conditional failure probabilities, and updates the empirical CDF accordingly. Within a unified least squares estimation framework, an ICP-CDF-LS parameter estimation method is developed, in which both the CDF and distribution parameters are updated iteratively. Simulation studies and case analyses demonstrate that, compared with the LS and MLE methods, the proposed approach achieves superior overall performance in terms of estimation accuracy and stability, making it more suitable for practical engineering applications.