math.ST — Statistics Theory

Nearly Minimax Variance Estimation Under Rough Random Design

Contributed by P. M. Aronow, Patrick Lopatto

We determine, up to a power of $\log n$, the minimax risk 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, for all sufficiently large $n$, between $c\Psi_n$ and $C\Psi_n(\log n)^{\Gamma}$, where $\Psi_n=n^{-2(s+1)/(d+4)}e^{-\kappa\sqrt{\log n}}(\log n)^{(s-1)/(d+4)}$ and the constants $\kappa>0$ and $\Gamma>0$ are explicit. In particular, the minimax exponent is $2(s+1)/(d+4)$, the minimax risk is smaller than $n^{-2(s+1)/(d+4)}$ by a stretched-exponential factor whose constant $\kappa$ is identified, and the rate proposed by Robins is not uniformly attainable over this model class. For $0<s\le1$, we show that the exact minimax rate is $n^{-1/2}\vee n^{-4s/(d+4s)}$; for $s>1$ and $d\le4s$, it is $n^{-1/2}$.