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Parameter estimation and optimal experimental design problems have been widely studied across scienceand engineering. The two are inextricably linked, with optimally designed experiments leading to better-estimated parameters. This link becomes even more crucial when available experiments produce minimal data due to practical constraints of limited experimental budgets. This work presents a novel framework that allows for the identification of optimal experimental arrangement, from a finite set of possibilities, for precise parameter estimation. The proposed framework relies on two pillars. First, we use multi-fidelity modeling to create reliable surrogates that relate unknown parameters to a measurable quantity of interest for a multitude of available choices defined through a set of candidate control vectors. Secondly, we quantify the estimation potential of an arrangement from the set of control vectors through the examination of statistical sensitivity measures calculated from the constructed surrogates. The measures of sensitivity are defined using analysis of variance as well as directional statistics. Two numerical examples are provided, where we demonstrate how the correlation between the estimation potential and the frequency of precise parameter estimation can inform the choice of optimal arrangement.