Abstract: Asymptotically correct simultaneous confidence bands (SCBs) are proposed for the mean and variance functions of a nonparametric regression model based on deterministic designs. The variance estimation is as efficient, up to order 𝓃 ^-1/2, as an infeasible estimator if the mean function were known. Simulation experiments provide strong evidence that corroborates the asymptotic theory. The proposed SCBs are used to analyze two sets of strata pressure data from the Bullianta Coal Mine in Erdos City, Inner Mongolia, China.

Key words and phrases: Brownian motion, heteroscedasticity, kernel, oracle efficiency, spline, strata pressure.