Hadamard derivative

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.[1]

Definition

A map ${\displaystyle \phi$ :\,\mathbb {D} \to \mathbb {E} } between Banach spaces ${\displaystyle \mathbb {D} }$ and ${\displaystyle \mathbb {E} }$ is Hadamard directionally differentiable[2] at ${\displaystyle \theta \in \mathbb {D} }$ in the direction ${\displaystyle h\in \mathbb {D} }$ if there exists a map ${\displaystyle \phi _{\theta }':\,\mathbb {D} \to \mathbb {E} }$ such that${\displaystyle {\frac {\phi (\theta +t_{n}h_{n})-\phi (\theta )}{t_{n}}}\to \phi _{\theta }'(h)}$ for all sequences ${\displaystyle h_{n}\to h}$ and ${\displaystyle t_{n}\downarrow 0}$. Note that this definition does not require continuity or linearity of the derivative with respect to the direction ${\displaystyle h}$. Although continuity follows automatically from the definition, linearity does not.

Relation to other derivatives

• If Hadamard directional derivative exists, then Gateaux derivative also exists and the two derivatives coincide[2]
• 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 ${\displaystyle X_{n}}$ be a sequence of random elements in a Banach space ${\displaystyle \mathbb {D} }$ (equipped with Borel sigma-field) such that weak convergence ${\displaystyle \tau _{n}(X_{n}-\mu )\to Z}$ holds for some ${\displaystyle \mu \in \mathbb {D} }$, some sequence of real numbers ${\displaystyle \tau _{n}\to \infty }$ and some random element ${\displaystyle Z\in \mathbb {D} }$ with values concentrated on a separable subset of ${\displaystyle \mathbb {D} }$. Then for a measurable map ${\displaystyle \phi$ :\mathbb {D} \to \mathbb {E} } that is Hadamard directionally differentiable at ${\displaystyle \mu }$ we have ${\displaystyle \tau _{n}(\phi (X_{n})-\phi (\mu ))\to \phi _{\mu }'(Z)}$ (where the weak convergence is with respect to Borel sigma-field on the Banach space ${\displaystyle \mathbb {E} }$).

This result has applications in optimal inference for wide range of econometric models, including models with partial identification and weak instruments.[3]

References

1. Shapiro, Alexander (1990). "On concepts of directional differentiability". Journal of Optimization Theory and Applications. 66 (3): 477–487. CiteSeerX 10.1.1.298.9112. doi:10.1007/bf00940933.
2. Shapiro, Alexander (1991). "Asymptotic analysis of stochastic programs". Annals of Operations Research. 30 (1): 169–186. doi:10.1007/bf02204815.
3. Fang, Zheng; Santos, Andres (2014). "Inference on directionally differentiable functions". arXiv:1404.3763 [math.ST].