Degradation analysis is essential in system health management and remaining useful life prediction. Since the observed degradation data are inevitably contaminated by measurement error, degradation state estimation is hence important for a more accurate evaluation of the health status. There are two challenges for estimating the degradation state. The first is the uncertainty associated with the estimated parameters for the model, and the other is the measurement outlier. Current models usually assume Gaussian measurement errors and they are sensitive to the measurement outlier. To deal with these two challenges, we develop a framework for degradation state estimation under the context of the distributionally robust optimization, which is robust to the parameter uncertainty. We further incorporate the Huber loss into this framework to make it robust to the measurement outlier. A procedure for estimation of the model parameters as well as setting the parameters of the ambiguity set is provided. The effectiveness of the model is validated using numerical and real case studies.