Dualizing Le Cam’s method for functional estimation I: General theory

Le Cam’s method (or the two-point method) is a commonly used tool for obtaining statistical lower bound and especially popular for functional estimation problems. This work aims to explain and give conditions for the tightness of Le Cam’s lower bound in functional estimation from the perspective of convex duality. Under a variety of settings, it is shown that the maximization problem that searches for the best two-point lower bound, upon dualizing, becomes a minimization problem minimizing an upper bound on the quadratic risk over a family of estimators. Since by the minimax theorem two problems have the same value, this value also characterizes (up to a universal factor) the optimal estimation rate. For estimating linear functionals of a distribution, our work strengthens prior results of Donoho–Liu (Ann. Statist. 19 (1991) 633–667) (for quadratic loss) by dropping the Hölderian assumption on the modulus of continuity. For exponential families, our results extend those of Juditsky–Nemirovski (Ann. Statist. 37 (2009) 2278–2300) by characterizing the minimax risk for the quadratic loss under weaker assumptions on the exponential family. We also provide an extension to the high-dimensional setting for estimating separable functionals. An application of our methodology to the area of “estimating the unseens” is provided in the companion paper (Polyanskiy and Wu (2023)), resolving the optimal rates (within logarithmic factors) of the distinct elements problem and Fisher’s species problem.