Non-Bayesian Post-Model-Selection Estimation as Estimation Under Model Misspecification

TL;DR

This paper explores how to perform parameter estimation after model selection when the true model is unknown, proposing new estimators and bounds that account for the selection process and outperform traditional bounds.

Contribution

It introduces three interpretations of post-model-selection estimation as model misspecification, developing corresponding estimators and bounds for each, enhancing understanding of their properties and performance.

Findings

Proposed estimators outperform traditional bounds in simulations.

The selective inference interpretation yields the lowest MSE among the methods.

Performance bounds are more informative than the oracle CRB.

Abstract

In many parameter estimation problems, the exact model is unknown and is assumed to belong to a set of candidate models. In such cases, a predetermined data-based selection rule selects a parametric model from a set of candidates before the parameter estimation. The existing framework for estimation under model misspecification does not account for the selection process that led to the misspecified model. Moreover, in post-model-selection estimation, there are multiple candidate models chosen based on the observations, making the interpretation of the assumed model in the misspecified setting non-trivial. In this work, we present three interpretations to address the problem of non-Bayesian post-model-selection estimation as an estimation under model misspecification problem: the naive interpretation, the normalized interpretation, and the selective inference interpretation, and discuss their properties. For each of these interpretations, we developed the corresponding misspecified maximum likelihood estimator and the misspecified Cram$\acute{\text{e}}$r-Rao-type lower bound. The relations between the estimators and the performance bounds, as well as their properties, are discussed. Finally, we demonstrate the performance of the proposed estimators and bounds via simulations of estimation after channel selection. We show that the proposed performance bounds are more informative than the oracle Cram$\acute{\text{e}}$r-Rao Bound (CRB), where the third interpretation (selective inference) results in the lowest mean-squared-error (MSE) among the estimators.