Rate adaptive estimation of the center of a symmetric distribution
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Given univariate random variables Y_1, …, Y_n with the Uniform(θ_0 - 1, θ_0 + 1) distribution, the sample midrange Y_(n)+Y_(1)/2 is the MLE for θ_0 and estimates θ_0 with error of order 1/n, which is much smaller compared with the 1/√(n) error rate of the usual sample mean estimator. However, the sample midrange performs poorly when the data has say the Gaussian N(θ_0, 1) distribution, with an error rate of 1/√(log n). In this paper, we propose an estimator of the location θ_0 with a rate of convergence that can, in many settings, adapt to the underlying distribution which we assume to be symmetric around θ_0 but is otherwise unknown. When the underlying distribution is compactly supported, we show that our estimator attains a rate of convergence of n^-1/α up to polylog factors, where the rate parameter α can take on any value in (0, 2] and depends on the moments of the underlying distribution. Our estimator is formed by the ℓ^γ-center of the data, for a γ≥2 chosen in a data-driven way – by minimizing a criterion motivated by the asymptotic variance. Our approach can be directly applied to the regression setting where θ_0 is a function of observed features and motivates the use of ℓ^γ loss function for γ > 2 in certain settings.
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