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We revisit the problem of estimating the mean of a real-valued distribution, presenting a novel estimator with sub-Gaussian convergence: intuitively, our estimator, on any distribution, is as accurate as the sample mean is for the Gaussian distribution of matching variance. Crucially, in contrast to prior works, our estimator does not require prior knowledge of the variance, and works across the entire gamut of distributions with bounded variance, including those without any higher moments. Parameterized by the sample size $n$, the failure probability $delta$, and the variance $sigma^2$, our estimator is accurate to within $sigmacdot(1+o(1))sqrt{frac{2logfrac{1}{delta}}{n}}$, tight up to the $1+o(1)$ factor. Our estimator construction and analysis gives a framework generalizable to other problems, tightly analyzing a sum of dependent random variables by viewing the sum implicitly as a 2-parameter $psi$-estimator, and constructing bounds using mathematical programming and duality techniques.
We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a non-asymptotic point of view. In particular, we define estimators with a sub-Gaussia
Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (suc
We consider the problem of estimating the covariance matrix of a random signal observed through unknown translations (modeled by cyclic shifts) and corrupted by noise. Solving this problem allows to discover low-rank structures masked by the existenc
We study the problem of outlier robust high-dimensional mean estimation under a finite covariance assumption, and more broadly under finite low-degree moment assumptions. We consider a standard stability condition from the recent robust statistics li
We study the least squares estimator in the residual variance estimation context. We show that the mean squared differences of paired observations are asymptotically normally distributed. We further establish that, by regressing the mean squared diff