High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors
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In location estimation, we are given n samples from a known distribution f shifted by an unknown translation λ, and want to estimate λ as precisely as possible. Asymptotically, the maximum likelihood estimate achieves the Cramér-Rao bound of error 𝒩(0, 1/nℐ), where ℐ is the Fisher information of f. However, the n required for convergence depends on f, and may be arbitrarily large. We build on the theory using smoothed estimators to bound the error for finite n in terms of ℐ_r, the Fisher information of the r-smoothed distribution. As n →∞, r → 0 at an explicit rate and this converges to the Cramér-Rao bound. We (1) improve the prior work for 1-dimensional f to converge for constant failure probability in addition to high probability, and (2) extend the theory to high-dimensional distributions. In the process, we prove a new bound on the norm of a high-dimensional random variable whose 1-dimensional projections are subgamma, which may be of independent interest.
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