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 $\phi :\mathbb{𝔻}\to \mathbb{𝔼}$ between Banach spaces $\mathbb{𝔻}$ and $\mathbb{𝔼}$ is Hadamard-directionally differentiable at $\theta \in \mathbb{𝔻}$ in the direction $h\in \mathbb{𝔻}$ if there exists a map ${{\phi }_{\theta }}^{\prime }:\mathbb{𝔻}\to \mathbb{𝔼}$ 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 $\mathbb{𝔻}$ (equipped with Borel sigma-field) such that weak convergence ${\tau }_n(X_n-\mu )\to Z$ holds for some $\mu \in \mathbb{𝔻}$, some sequence of real numbers ${\tau }_n\to \mathrm{\infty }$ and some random element $Z\in \mathbb{𝔻}$ with values concentrated on a separable subset of $\mathbb{𝔻}$. Then for a measurable map $\phi :\mathbb{𝔻}\to \mathbb{𝔼}$ 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 $\mathbb{𝔼}$).

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

See also

• Directional derivative – Instantaneous rate of change of the function
• Fréchet derivative – Derivative defined on normed spaces - generalization of the total derivative
• Gateaux derivative – Generalization of the concept of directional derivative
• Generalizations of the derivative – Fundamental construction of differential calculus
• Total derivative – Type of derivative in mathematics