For nonparametric regression estimation on a bounded interval, optimal rates of decrease for integrated mean square error are known but not the best possible constants. A sharp result on such a constant, i.e., an analog of Fisher's bound for asymptotic variances is obtained for minimax risk over a Sobolev smoothness class. Normality of errors is assumed. The method is based on applying a recent result on minimax filtering in Hilbert space. A variant of spline smoothing is developed to deal with noncircular models.
"Spline Smoothing in Regression Models and Asymptotic Efficiency in $L_2$." Ann. Statist. 13 (3) 984 - 997, September, 1985. https://doi.org/10.1214/aos/1176349651