Post-Parameter-Selection Maximum-Likelihood Estimation

Practical multi-parameter estimation problems with limited measurements and unidentifiable scenarios usually involve a preliminary data-based selection stage. Recent works have shown that selection-aware estimation methods outperform state-of-the-art estimators in the sense of post-selection bias and post-selection mean-squared-error (PSMSE). In this paper, we discuss non-Bayesian estimation methods where a subset of parameters is selected for estimation from the full unknown parameter vector by a data-based selection-rule. We present four estimators: the maximum likelihood (ML), the coherent ML, the post-selection ML (PSML), and the coherent PSML. Coherent post-selection estimators force the unselected parameters to zero, and thus, can be implemented in practical high-dimensional problems. Additionally, we develop a low-complexity algorithm, the stochastic approximation PSML (SA-PSML) for practical implementation of the coherent PSML estimator. Simulation results show that the SA-PSML algorithm achieves a lower PSMSE than the coherent ML estimator for sparse vector recovery with the orthogonal matching pursuit (OMP) selection rule.