Hadamard derivative

In mathematics, the Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications in stochastic programming and asymptotic statistics.

Definition

A map between Banach spaces and is Hadamard-directionally differentiable at in the direction if there exists a map such that
for all sequences and.
Note that this definition does not require continuity or linearity of the derivative with respect to the direction. Although continuity follows automatically from the definition, linearity does not.

Relation to other derivatives

If the Hadamard directional derivative exists, then the Gateaux derivative also exists and the two derivatives coincide.
The Hadamard derivative is readily generalized for maps between Hausdorff topological vector spaces.

Applications

A version of functional delta method holds for Hadamard directionally differentiable maps. Namely, let be a sequence of random elements in a Banach space such that weak convergence holds for some, some sequence of real numbers and some random element with values concentrated on a separable subset of. Then for a measurable map that is Hadamard directionally differentiable at we have .
This result has applications in optimal inference for wide range of econometric models, including models with partial identification and weak instruments.