Bias Reduction of MLEs for Weibull Distributions under Grouped Lifetime Data

Abstract

Accelerated life tests (ALTs) usually contain subsampling because of the cost-effective evaluation. The two-stage method can deal with grouped data with subsampling, but the bias of maximum likelihood estimates (MLEs) can be alarmingly high. In this article, we propose reducing the bias of MLEs for grouped data via an unbiasing factor method. We introduce the procedures and give the unbiasing factor values. The proposed method is studied and compared with the modified maximum likelihood method for relative bias (RB) and mean square error (MSE) in Monte Carlo simulations. The results show that the unbiasing factor method is better in most cases.

Bootstrapping Analysis of Lifetime Data with Subsampling

“…Table II shows that the values of log ηˆi obtained by the proposed method are larger than the results of Freeman and Vining's method, indicating that the maximum likelihood method underestimated log ηˆi. 15 The unbiased estimate of the shape parameter is βˆU = 2.7160, which is smaller than the Freeman and Vining's shape parameter estimate, βˆF V = 3.62. From Table II, we can see that there is a large difference between the c var log ηˆi ð Þ obtained by two methods.…”

“…It can be easily proved that the ratio between the estimator and the shape parameter obtained from Yang and Lin is a pivotal quantity for complete and type II censoring, which can reduce the bias further via an unbiasing factor method. Accordingly, Wang et al . proposed reducing the biases of parameters through an unbiasing factor method; and their simulations showed that the unbiasing factor method developed is better than Yang and Lin's method for complete or censored data.…”

“…The theoretical justification for using the unbiasing factor method is based on the existence of pivotal quantity. Wang et al . proved that and are pivotal quantities for complete and type II censoring under grouped data.…”

“…The theoretical justification for using the unbiasing factor method is based on the existence of pivotal quantity. Wang et al 15 proved that βˆ=β and β log ηˆi=η i ð Þare pivotal quantities for complete and type II censoring under grouped data. They are independent of the true values of β and η i but dependent on the number of stress levels m, sample size n i , and the number of failures r i .…”

“…Theoretically, we cannot reduce the bias via an unbiasing factor method. Wang et al . showed that the nearly unbiasing factors can be calculated by replacing r i with n i in and .…”