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 $$\varphi : \mathbb{D}\to \mathbb{E}$$ 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 $$\varphi_\theta': \, \mathbb{D} \to \mathbb{E}$$ such that $$\frac{\varphi(\theta+t_n h_n)-\varphi(\theta)}{t_n} \to \varphi_\theta'(h)$$ 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 $$\mathbb{D}$$ (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 $$\tau_n\to \infty$$ and some random element $$Z \in \mathbb{D}$$ with values concentrated on a separable subset of $$\mathbb{D}$$. Then for a measurable map $$\varphi: \mathbb{D}\to\mathbb{E}$$ 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 $$\mathbb{E}$$).

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