Parameter Estimation Methods for Condition-Based Maintenance With Indirect Observations

This article proposes methods to estimate the parameters of condition monitored equipment whose failure rate follows the Cox's time-dependent Proportional Hazards Model. Due to errors of measurement, of interpretation, or due to limited accuracy of measurement instruments, the observation process is not perfect, and does not directly reveal the exact degradation state. At each observation moment, we observe and collect information about an indicator of the underlying unobservable degradation state. To match the indicator's value to the unobservable degradation state, the stochastic relation between them is given by an observation probability matrix. In this study, we consider the case of imperfect observations, and we assume that the equipment's unobservable degradation state transition follows a Hidden Markov Model. We determine the Probability Density Function of the time to failure, and use the Maximum Likelihood Estimation to estimate the model's parameters. These are the Proportional Hazards', and the Hidden Markov Models' parameters. We study the cases of censored, and uncensored data; and carry out simulation studies to test the accuracy, and the convergence of the estimation methods.