Nearly-minimax variance estimation under rough random design

Abstract

We identify the minimax exponent for constant conditional variance estimation under rough random design. The unknown design density is bounded above and away from zero, with no smoothness assumption, and the conditional error laws may depend on the covariates and have uniformly bounded fourth moments. For an $s$-Hölder regression function with $s>1$ in dimension $d>4s$, the minimax root-mean-square risk lies between $c{n}^{-2(s+1)/(d+4)}{e}^{-C\sqrt{\log n}}$ and $C{n}^{-2(s+1)/(d+4)}$. These bounds show that the rate proposed by Robins is not uniformly attainable over this model class. For $0<s\le 1$, we show the exact minimax rate is $n^{-1/2}\vee n^{-4s/(d+4s)}$; for $s>1$ and $d\le 4s$, it is $n^{-1/2}$.

78 pages