Pointwise minimax rate for Lipschitz regression

At an interior target point in regular fixed-design nonparametric regression, mean functions with Lipschitz bound $L$ have pointwise minimax risk of order $(L{\sigma }^2/n{)}^{2/3}$ for fixed positive $L,\sigma$ as $n\to \infty$. A smoothing bandwidth of order $({\sigma }^2/(n{L}^2){)}^{1/3}$ balances squared bias of an estimator and variance; the two-point lower bound with a triangular bump gives the matching lower rate.