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Given $n$ samples from a population of individuals belonging to different types with unknown proportions, how do we estimate the probability of discovering a new type at the $(n+1)$-th draw? This is a classical problem in statistics, commonly referred to as the missing mass estimation problem. Recent results by Ohannessian and Dahleh \citet{Oha12} and Mossel and Ohannessian \citet{Mos15} showed: i) the impossibility of estimating (learning) the missing mass without imposing further structural assumptions on the type proportions; ii) the consistency of the Good-Turing estimator for the missing mass under the assumption that the tail of the type proportions decays to zero as a regularly varying function with parameter $\alpha\in(0,1)$. In this paper we rely on tools from Bayesian nonparametrics to provide an alternative, and simpler, proof of the impossibility of a distribution-free estimation of the missing mass. Up to our knowledge, the use of Bayesian ideas to study large sample asymptotics for the missing mass is new, and it could be of independent interest. Still relying on Bayesian nonparametric tools, we then show that under regularly varying type proportions the convergence rate of the Good-Turing estimator is the best rate that any estimator can achieve, up to a slowly varying function, and that minimax rate must be at least $n^{-\alpha/2}$. We conclude with a discussion of our results, and by conjecturing that the Good-Turing estimator is an rate optimal minimax estimator under regularly varying type proportions.

Purpose\ud\udTo develop a framework to fully characterize quantitative magnetization transfer indices in the human cervical cord in vivo within a clinically feasible time.\udMethods\ud\udA dedicated spinal cord imaging protocol for quantitative magnetization transfer was developed using a reduced field‐of‐view approach with echo planar imaging (EPI) readout. Sequence parameters were optimized based in the Cramer‐Rao‐lower bound. Quantitative model parameters (i.e., bound pool fraction, free and bound pool transverse relaxation times [ urn:x-wiley:07403194:media:mrm26909:mrm26909-math-0014, urn:x-wiley:07403194:media:mrm26909:mrm26909-math-0015], and forward exchange rate [kFB]) were estimated implementing a numerical model capable of dealing with the novelties of the sequence adopted. The framework was tested on five healthy subjects.\udResults\ud\udCramer‐Rao‐lower bound minimization produces optimal sampling schemes without requiring the establishment of a steady‐state MT effect. The proposed framework allows quantitative voxel‐wise estimation of model parameters at the resolution typically used for spinal cord imaging (i.e. 0.75 × 0.75 × 5 mm3), with a protocol duration of ∼35 min. Quantitative magnetization transfer parametric maps agree with literature values. Whole‐cord mean values are: bound pool fraction = 0.11(±0.01), urn:x-wiley:07403194:media:mrm26909:mrm26909-math-0016 = 46.5(±1.6) ms, urn:x-wiley:07403194:media:mrm26909:mrm26909-math-0017 = 11.0(±0.2) µs, and kFB = 1.95(±0.06) Hz. Protocol optimization has a beneficial effect on reproducibility, especially for urn:x-wiley:07403194:media:mrm26909:mrm26909-math-0018 and kFB.\udConclusion\ud\udThe framework developed enables robust characterization of spinal cord microstructure in vivo using qMT. Magn Reson Med 79:2576–2588, 2018. © 2017 The Authors Magnetic Resonance in Medicine published by Wiley Periodicals, Inc. on behalf of International Society for Magnetic Resonance in Medicine. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.