Estimation after Parameter Selection: Performance Analysis and Estimation Methods

TL;DR

This paper analyzes the impact of parameter selection on estimation accuracy, introduces a performance measure and bounds, and proposes an estimator tailored for post-selection scenarios, validated through various distribution examples.

Contribution

It introduces a post-selection mean squared error criterion, derives a Cramér-Rao-type bound, and develops a post-selection maximum-likelihood estimator with iterative implementation methods.

Findings

The PSML estimator achieves the Cramér-Rao bound under certain conditions.

The proposed bounds and estimators are validated with different distributions.

Post-selection bias significantly affects estimation performance and can be mitigated by the proposed methods.

Abstract

In many practical parameter estimation problems, prescreening and parameter selection are performed prior to estimation. In this paper, we consider the problem of estimating a preselected unknown deterministic parameter chosen from a parameter set based on observations according to a predetermined selection rule, $\Psi$. The data-based parameter selection process may impact the subsequent estimation by introducing a selection bias and creating coupling between decoupled parameters. This paper introduces a post-selection mean squared error (PSMSE) criterion as a performance measure. A corresponding Cram\'er-Rao-type bound on the PSMSE of any $\Psi$-unbiased estimator is derived, where the $\Psi$-unbiasedness is in the Lehmann-unbiasedness sense. The post-selection maximum-likelihood (PSML) estimator is presented .It is proved that if there exists an $\Psi$-unbiased estimator that achieves the $\Psi$-Cram\'er-Rao bound (CRB), i.e. an $\Psi$-efficient estimator, then it is produced by the PSML estimator. In addition, iterative methods are developed for the practical implementation of the PSML estimator. Finally, the proposed $\Psi$-CRB and PSML estimator are examined in estimation after parameter selection with different distributions.