Nearly minimax variance estimation under rough random design
Minimax root-mean-square bounds for estimating a constant conditional variance from independent random-design regression observations. The design density has fixed positive lower and upper bounds but no smoothness; covariate-dependent errors have uniformly bounded conditional fourth moments. For 1 < s < d/4, the proved bracket identifies the polynomial exponent 2(s+1)/(d+4) and the common explicit stretched-exponential factor, with a remaining logarithmic gap. It also proves the exponent corollary, the exact low-smoothness rate, and the parametric regime.
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