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.

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