Sharp Optimality of Simple, Plug-in Estimation of the Fisher Information of a Smoothed Density

Given independent and identically distributed data from a compactly supported, α-Hölder density f , we study estimation of the Fisher information of the Gaussian-smoothed density f * φ t , where φ t is the density of N (0, t). We derive the minimax rate including the sharp dependence on t and show some simple, plug-in type estimators are optimal for t > 0, even though extra debiasing steps are widely employed in the literature to achieve the sharp rate in the unsmoothed (t = 0) case. Due to our result's sharp characterization of the scaling in t, plug-in estimators of the mutual information and entropy are shown to achieve the parametric rate by way of the I-MMSE and de Bruijn's identities.