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 $\phi :𝔻\to 𝔼$ between Banach spaces $𝔻$ and $𝔼$ is Hadamard-directionally differentiable at $\theta \in 𝔻$ in the direction $h\in 𝔻$ if there exists a map ${{\phi }_{\theta }}^{\prime }:𝔻\to 𝔼$ such that $$\frac{\phi (\theta +t_n{h}_n)-\phi (\theta )}{t_n}\to {{\phi }_{\theta }}^{\prime }(h)$$ for all sequences $h_n\to h$ and $t_n\to 0$.

Note that this definition does not require continuity or linearity of the derivative with respect to the direction $h$. 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.
• 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 $X_n$ be a sequence of random elements in a Banach space $𝔻$ (equipped with Borel sigma-field) such that weak convergence ${\tau }_n(X_n-\mu )\to Z$ holds for some $\mu \in 𝔻$, some sequence of real numbers ${\tau }_n\to \mathrm{\infty }$ and some random element $Z\in 𝔻$ with values concentrated on a separable subset of $𝔻$. Then for a measurable map $\phi :𝔻\to 𝔼$ that is Hadamard directionally differentiable at $\mu$ we have ${\tau }_n(\phi (X_n)-\phi (\mu ))\to {{\phi }_{\mu }}^{\prime }(Z)$ (where the weak convergence is with respect to Borel sigma-field on the Banach space $𝔼$).

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

See also

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• Template:Annotated link - generalization of the total derivative
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