Hadamard derivative

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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.^[1]

Definition[]

A map ${\displaystyle \varphi :\mathbb {D} \to \mathbb {E} }$ 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 ${\displaystyle \varphi _{\theta }':\,\mathbb {D} \to \mathbb {E} }$ such that${\displaystyle {\frac {\varphi (\theta +t_{n}h_{n})-\varphi (\theta )}{t_{n}}}\to \varphi _{\theta }'(h)}$ 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 the Hadamard directional derivative exists, then the Gateaux derivative also exists and the two derivatives coincide.^[2]
• 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 ${\displaystyle X_{n}}$ 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 ${\displaystyle \varphi :\mathbb {D} \to \mathbb {E} }$ that is Hadamard directionally differentiable at ${\displaystyle \mu }$ we have ${\displaystyle \tau _{n}(\varphi (X_{n})-\varphi (\mu ))\to \varphi _{\mu }'(Z)}$ (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[]