On Differentiable Functionals

Given a sample of size n from a distribution P_λ, one wants to estimate a functional ψ(λ) of the (typically infinite-dimensional) parameter λ. Lower bounds on the performance of estimators can be based on the concept of a differentiable functional P_λ → ψ(λ). In this paper we relate a suitable definition of differentiable functional to differentiability of α → dP1/2_λ and λ → ψ(λ). Moreover, we show that regular estimability of a functional implies its differentiability.