Robust Maximum Likelihood Estimation

Abstract

In many applications, statistical estimators serve to derive conclusions from data, for example, in finance, medical decision making, and clinical trials. However, the conclusions are typically dependent on uncertainties in the data. We use robust optimization principles to provide robust maximum likelihood estimators that are protected against data errors. Both types of input data errors are considered: (a) the adversarial type, modeled using the notion of uncertainty sets, and (b) the probabilistic type, modeled by distributions. We provide efficient local and global search algorithms to compute the robust estimators and discuss them in detail for the case of multivariate normally distributed data. The estimator performance is demonstrated on two applications. First, using computer simulations, we demonstrate that the proposed estimators are robust against both types of data uncertainty and provide more accurate estimates compared with classical estimators, which degrade significantly, when errors are encountered. We establish a range of uncertainty sizes for which robust estimators are superior. Second, we analyze deviations in cancer radiation therapy planning. Uncertainties among plans are caused by patients’ individual anatomies and the trial-and-error nature of the process. When analyzing a large set of past clinical treatment data, robust estimators lead to more reliable decisions when applied to a large set of past treatment plans.