Advanced Bayesian Estimation of Weibull Early Life Failure Distributions

In semiconductor manufacturing, it is a key to ensure reliability of the produced devices. The population's reliability level is demonstrated by means of a burn-in study (that is investigating a large number of devices under real-life stress conditions for product relevant fails). Burn-in settings are based on the lifetime distribution of early fails. Typically, it is modelled as a Weibull distribution Wb(a,b) with scale parameter a > 0 and shape parameter b is an element of (0,1) motivated by a decreasing failure rate within the devices' early life. Depending on the applied burn-in scheme, the Weibull parameters have to be estimated from time-to-failure and discrete failure count data, respectively. In this paper, we present advanced Bayesian estimation models for the Weibull distribution handling both data situations. First, a simplified conjugate approach using gamma-histogram-beta priors is presented. Further, according to the paper's main focus, an extended Bayesian concept for assessing Weibull early life failure distributions is highlighted. It is characterized by a Dirichlet prior distribution applied to the lifetime function of early fails. The proposed model simplifies the incorporation of engineering prior knowledge. Moreover, it can be extended to both discrete failure and time-to-failure burn-in data. The joint posterior distribution, Bayesian estimators and compounded and joint credible regions are derived by means of Monte Carlo simulation. The principle of Bayesian learning allows to update the Weibull early life failure distribution whenever new failure data become available. Therefore, burn-in settings can dynamically be adapted improving the efficiency of burn-in. Copyright (c) 2014 John Wiley & Sons, Ltd.