Performance prediction of implicitly defined estimators of non-random parameters

Mehmetcik, Erdal
This thesis study is concerned with performance prediction for estimators with non-random parameters. A rather general class of estimators, called implicitly defined estimators (IDEs), is of main interest. An implicitly defined estimator declares the minimizer/maximizer of a selected cost/reward function as the parameter estimate. The maximum likelihood (ML) and the least squares estimators are among the well-known examples of this class. An exact MSE expression for implicitly defined estimators with a symmetric and unimodal objective function is derived. It is also shown that the expression reduces to the Cramer-Rao lower bound (CRLB) and mis-specified CRLB in the large sample size regime for ML and mis-specified ML estimation, respectively. The expression is shown to yield the Ziv-Zakai bound (without the valley filling function) for the maximum a posteriori (MAP) estimator when it is used in a Bayesian setting, that is, when an a-priori distribution is assigned to the unknown parameter. Extension of the suggested expression to the case of nuisance parameters is studied and some approximations are given to ease the computations for this case. Numerical results indicate that the suggested MSE expression not only predicts the estimator performance in the asymptotic region; but it is also applicable for the threshold region analysis, even for IDEs whose objective functions do not satisfy the symmetry and unimodality assumptions. Advantages of the suggested MSE expression are its conceptual simplicity and its relatively straightforward numerical calculation due to the reduction of the estimation problem to a binary hypothesis testing problem, similar to the usage of Ziv-Zakai bounds in random parameter estimation problems. The proposed approach for MSE approximation is adapted for bias prediction applications, and similar numerical studies are repeated. Several possible applications of the proposed performance prediction method are studied, and example cases are given.


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Citation Formats
E. Mehmetcik, “Performance prediction of implicitly defined estimators of non-random parameters,” Ph.D. - Doctoral Program, Middle East Technical University, 2023.