On Differentiable Functionals

Abstract

Given a sample of size $n$ from a distribution $P_\lambda$, one wants to estimate a functional $\psi(\lambda)$ of the (typically infinite-dimensional) parameter $\lambda$. Lower bounds on the performance of estimators can be based on the concept of a differentiable functional $P_\lambda \rightarrow \psi(\lambda)$. In this paper we relate a suitable definition of differentiable functional to differentiability of $\alpha \rightarrow dP^{1/2}_\lambda$ and $\lambda \rightarrow \psi(\lambda)$. Moreover, we show that regular estimability of a functional implies its differentiability.