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.
Description
Estimation after Parameter Selection: Performance Analysis and
Estimation Methods
%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
}