This paper compares three methods for producing lower bounds on the minimax risk under quadratic loss. The first uses the bounds from Brown and Gajek. The second method also uses the information inequality and results in bounds which are always at least as good as those from the first method. The third method is the hardest-linear-family method described by Donoho and Liu. These methods are applied in four examples, the last of which relates to a frequently considered problem in nonparametric regression.
"Information Inequality Bounds on the Minimax Risk (with an Application to Nonparametric Regression)." Ann. Statist. 19 (1) 329 - 337, March, 1991. https://doi.org/10.1214/aos/1176347985