Estimation of lifetime distribution with covariates using online monitoring

Online monitoring becomes a common tool for keeping high reliability of products and systems in these days. The information to maintain the target product includes usage history, system conditions, and environmental conditions. Nowadays they are monitored and reported in real time and stored as big data. On modeling the failure mechanisms statistically, these variables are primary candidates for covariates which affect the failure mechanism. There is literature on modeling the lifetime of products with the accelerated lifetime model on an accumulation of the covariate effects, which is a nonlinear transformation of the observed lifetime. The existing literature requires a parametric form of the transformed lifetime distribution in advance. However, such knowledge may be difficult to acquire in advance in some cases. When we assume an incorrect distribution, it is called misspecification. This paper proposes a strategy to use the log-normal likelihood for the estimation of the parameters of the covariate effects when the underlying distribution is either the Weibull or log-normal distribution. It is derived that the score function of the log-normal likelihood is identified as an approximation for the Weibull cases. The simulation study shows the relationship among the sample distribution of parameter estimates and underlying distributions. For the Weibull cases, if the shape parameter is large, the bias of the resulting estimates is small.