Effects of Model Misspecification and Uncertainty on the Performance of Estimators

System designers across all disciplines of technology face the need to develop machines capable of independently processing and analyzing data and predicting future data. This is the fundamental problem of interest in ?estimation theory,? wherein probabilistic analyses are used to isolate relationships between variables, and in ?statistical inference,? wherein those variables are used to make inferences about real-world quantities. In practice, all estimators are designed based on limited statistical generalizations about the behavior of the observed and latent variables of interest; however, these models are rarely fully representative of reality. In such cases, there exists a ?model misspecification,? and the resulting estimators will produce results that differ from those of the properly specified estimators. Evaluating the performance of a given estimator may sometimes be done by direct comparison of estimator outputs to known ground truth. However, in many cases, there is no guarantee that future latent variables will follow the same rules as the limited ground truth available for analysis. When this is the case, it is useful to employ a second evaluation methodology: statistical generalization about estimator behavior based on candidate models of observed reality. The overall objective of this dissertation is to evaluate and expand upon state-of-the-art approaches to estimation and estimator analysis under various types of modeling errors. Four main contributions are provided in this area. First, we contribute a use-case demonstration of applying estimator performance bounds analysis in a practical application: improving estimator design by optimization of system parameters, applied to optimizing sensor trajectories in synthetic aperture direction of arrival (SA-DOA) based on the Cram?r-Rao lower bound (CRB). Next, we contribute an analysis of the efficacy of computationally efficient signal subspace algorithms (e.g. MUltiple SIgnal Classification (MUSIC)) for performing multi-antenna radio direction finding in the presence of various modeling errors (such as antennae mutual coupling or sensor placement errors). To that aim, we leverage the framework of misspecified Cram?r-Rao lower bound (MCRB) to characterize the impact of modeling errors. This thesis provides an analysis that extends the known results on the equivalence of maximum-likelihood estimation (MLE) and MUSIC methods to cases of model misspecification. Next, we contribute an online method of Bayesian covariance estimation that leverages conjugate prior analysis, which, when embedded within Kalman-type filtering architectures, may be used to adapt to real-time changes in sensor performance. Finally, we focus on the analysis of state estimation problems under modeling. In those situations, we show that the expected value of a state estimator converges to the so-called pseudotrue state, which is formalized in this thesis for the first time. This result has important implications in defining theoretical estimation bounds under model misspecification in filtering contexts, for instance the misspecified posterior Cram?r-Rao lower bound (MPCRB) which is applicable to practical dynamical estimation problems, and solutions, including the popular Kalman filter and other similar estimators. Particularization of the proposed pseudotrue expression to linear/Gaussian systems results in a form which may be cheaply computed for an arbitrary set of hypothetical ground truth states, allowing for rapid analysis of dynamic estimator performance against any number of proposed ground truth models.