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.

翻译：在许多参数估计问题中，精确模型未知且被假定属于一组候选模型。在此类情况下，预先设定的基于数据的选取规则会在参数估计前从候选模型中选定一个参数模型。现有的模型误设估计框架并未考虑导致误设模型的选择过程。此外，在模型选择后估计中，多个候选模型是基于观测结果选出的，这使得在误设情形下对所假设模型的解释变得不平凡。本文提出三种解释方法，将非贝叶斯模型选择后估计问题转化为模型误设下的估计问题：朴素解释、归一化解释和选择性推断解释，并讨论其性质。针对每种解释，我们开发了相应的误设最大似然估计器及误设克莱默-拉奥型下界。讨论了估计器与性能界之间的关系及其性质。最后，通过信道选择后的参数估计仿真验证了所提估计器与性能界的表现。结果表明，所提性能界比奥拉克莱默-拉奥界（CRB）更具信息量，其中第三种解释（选择性推断）在估计器中实现了最低的均方误差（MSE）。