On Rates Attainable under Random Design: A Negative Answer to a Problem of Robins

We give a negative answer to a problem posed by James Robins on estimating a constant conditional variance in nonparametric regression under random design. For every $s>1$ and integer $d>4s$, when the regression function is $s$-H\"older, the unknown design density is bounded above and away from zero, and the conditional error laws may depend on the design but have mean zero, a common variance, and uniformly bounded fourth moments, we show that the minimax root-mean-square risk is bounded below by $n^{-\beta}$ with $\beta=\frac{d(3s+1)+8s}{(d+2s)(d+4)}$. Hence the conjectured rate $n^{-4s/(d+4s)}$ is not uniformly attainable.