Estimation After Parameter Selection: Performance Analysis and Estimation Methods

Abstract

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, Ψ. 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 Cramér-Rao-type bound on the PSMSE of any Ψ-unbiased estimator is derived, where the Ψ-unbiasedness is in the Lehmann-unbiasedness sense. The post-selection maximumlikelihood (PSML) estimator is presented. It is proved that if there exists an Ψ-unbiased estimator that achieves the Ψ-Cramér-Rao bound (CRB), i.e., an Ψ-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 Ψ-CRB and PSML estimator are examined in estimation after parameter selection with different distributions.Index Terms-Non-Bayesian parameter estimation, Ψ-Cramér-Rao bound (Ψ-CRB), estimation after parameter selection, postselection maximum-likelihood (PSML) estimator, Lehmann unbiasedness.

Low-Complexity Methods for Estimation After Parameter Selection

“…The PSML estimator has been shown to have better performance than the ML estimator, in terms of Ψbias and PSMSE, in various scenarios [29], [30]. Moreover, it has been shown in [14], similarly to the ML estimator and the conventional efficiency, that if an Ψ-efficient estimator exists, then it coincides with the PSML estimator for the selected parameter. An Ψ-efficient estimator, as defined in Definition 3 in [14], is an Ψ-unbiased estimator that achieves the Ψ-CRB on the PSMSE, which is given in Section IV.…”

“…B) the maximization may require a multi-dimensional grid search, where the computational complexity increases with the dimension of θ. Hence, there is a need for practical, low-complexity estimation methods that use the special structure of the PSLL, as well as the tractability of the conventional log-likelihood part, f (x, y; θ), in order to approximate the solution for the score equation from (14).…”

“…In post-selection inference, it is well known that the selection stage has an impact on subsequent estimation, by creating coupling between parameters that originally were decoupled [14], leading to inaccurate confidence intervals, and introducing a selection bias [15]- [21]. The enlightening example by Efron [20, Fig.…”

“…In [14] we suggested the post-selection maximum likelihood (PSML) estimator for single-stage estimation after parameter selection and developed a novel Ψ-Cramér-Rao bound (CRB) on the post-selection mean squared error (PSMSE). arXiv:1904.02500v3 [eess.SP] 28 Jan 2020…”

“…This model is a generalization of the classical two-stage model for independent populations [17]. We derive the two-stage versions of the Ψ-unbiasedness in the Lehmann sense [32] and the PSML estimator, that extend our single-stage results in [14], [28]. In the main part of this paper, we develop practical, low-complexity, estimation methods for multivariate cases where the PSML estimator is intractable.…”