On Mean Estimation for Heteroscedastic Random Variables
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We study the problem of estimating the common mean μ of n independent symmetric random variables with different and unknown standard deviations σ_1 ≤σ_2 ≤⋯≤σ_n. We show that, under some mild regularity assumptions on the distribution, there is a fully adaptive estimator μ such that it is invariant to permutations of the elements of the sample and satisfies that, up to logarithmic factors, with high probability, |μ - μ| ≲min{σ_m^*, √(n)/∑_i = √(n)^n σ_i^-1} , where the index m^* ≲√(n) satisfies m^* ≈√(σ_m^*∑_i = m^*^nσ_i^-1).
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