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 $\varphi : \mathbb\to \mathbb$ between Banach spaces $\mathbb$ and $\mathbb$ is Hadamard-directionally differentiable at $\theta \in \mathbb$ in the direction $h \in \mathbb$ if there exists a map $\varphi_\theta': \, \mathbb \to \mathbb$ such that
$\frac \to \varphi_\theta'(h)$
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 $\mathbb$ (equipped with Borel sigma-field) such that weak convergence $\tau_n (X_n-\mu) \to Z$ holds for some $\mu \in \mathbb$, some sequence of real numbers $\tau_n\to \infty$ and some random element $Z \in \mathbb$ with values concentrated on a separable subset of $\mathbb$. Then for a measurable map $\varphi: \mathbb\to\mathbb$ that is Hadamard directionally differentiable at $\mu$ we have $\tau_n (\varphi(X_n)-\varphi(\mu)) \to \varphi_\mu'(Z)$ (where the weak convergence is with respect to Borel sigma-field on the Banach space $\mathbb$).

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

See also

*
* - generalization of the total derivative
*
*
*

References

{{Analysis in topological vector spaces

Directional statistics
Generalizations of the derivative