%0 Generic %1 routtenberg2015estimation %A Routtenberg, Tirza %A Tong, Lang %D 2015 %K cramer-rao machine-learning statistics %T Estimation after Parameter Selection: Performance Analysis and Estimation Methods %U http://arxiv.org/abs/1503.02045 %X 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 Cramér-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 $\Psi$-Cramér-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.

@misc{routtenberg2015estimation, 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.}, added-at = {2016-05-09T23:08:54.000+0200}, author = {Routtenberg, Tirza and Tong, Lang}, biburl = {https://www.bibsonomy.org/bibtex/2d206c416b722b28789118f222a103bab/shabbychef}, description = {Estimation after Parameter Selection: Performance Analysis and Estimation Methods}, interhash = {2e6c0ca849e074b57e72f5959a69ed42}, intrahash = {d206c416b722b28789118f222a103bab}, keywords = {cramer-rao machine-learning statistics}, note = {cite arxiv:1503.02045Comment: A submitted paper}, timestamp = {2016-05-09T23:08:54.000+0200}, title = {Estimation after Parameter Selection: Performance Analysis and Estimation Methods}, url = {http://arxiv.org/abs/1503.02045}, year = 2015 }