In many practical multiparameter estimation problems, no a-priori information exists regarding which parameters are more relevant within a group of candidate unknown parameters. This paper considers the estimation of a selected 'parameter of interest', where the selection is conducted according to a data-based selection rule, Ψ. The selection process introduces a selection bias and creates coupling between decoupled parameters. We propose a two-stage data-acquisition approach that can remove the selection bias and improve estimation performance. We derive a two-stage Cramér-Rao-type bound on the post-selection mean squared error (PSMSE) of any Ψ-unbiased estimator, where the Ψ-unbiasedness is in the Lehmann sense. In addition, we present the two-stage post-selection maximum-likelihood (PSML) estimator. The proposed Ψ-Cramer-Rao bound (CRB), PSML estimator and other existing estimators are examined for a linear Gaussian model, which is widely used in clinical research.