Lower bounds on the radius of various confidence sets for density functions and their derivatives are determined. In each case investigated, the smallest radius obtainable and the radius actually obtained by using known estimates exhibit the same dependency on the fixed sample size $n$. The lower bounds are derived using methods in Meyer (1976). Although only a few combinations of pseudometric and density class are considered here, the techniques illustrated can be used elsewhere with little conceptual difficulty.
"Bounds for Estimation of Density Functions and Their Derivatives." Ann. Statist. 5 (1) 136 - 142, January, 1977. https://doi.org/10.1214/aos/1176343746