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 φ : D → E {\displaystyle \varphi :\mathbb {D} \to \mathbb {E} } between Banach spaces D {\displaystyle \mathbb {D} } and E {\displaystyle \mathbb {E} } is Hadamard-directionally differentiable at θ ∈ D {\displaystyle \theta \in \mathbb {D} } in the direction h ∈ D {\displaystyle h\in \mathbb {D} } if there exists a map φ θ ′ : D → E {\displaystyle \varphi _{\theta }':\,\mathbb {D} \to \mathbb {E} } such that
φ ( θ + t n h n ) − φ ( θ ) t n → φ θ ′ ( h ) {\displaystyle {\frac {\varphi (\theta +t_{n}h_{n})-\varphi (\theta )}{t_{n}}}\to \varphi _{\theta }'(h)} for all sequences h n → h {\displaystyle h_{n}\to h} and t n → 0 {\displaystyle t_{n}\to 0} . Note that this definition does not require continuity or linearity of the derivative with respect to the direction h {\displaystyle 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 {\displaystyle X_{n}} be a sequence of random elements in a Banach space D {\displaystyle \mathbb {D} } (equipped with Borel sigma-field) such that weak convergence τ n ( X n − μ ) → Z {\displaystyle \tau _{n}(X_{n}-\mu )\to Z} holds for some μ ∈ D {\displaystyle \mu \in \mathbb {D} } , some sequence of real numbers τ n → ∞ {\displaystyle \tau _{n}\to \infty } and some random element Z ∈ D {\displaystyle Z\in \mathbb {D} } with values concentrated on a separable subset of D {\displaystyle \mathbb {D} } . Then for a measurable map φ : D → E {\displaystyle \varphi :\mathbb {D} \to \mathbb {E} } that is Hadamard directionally differentiable at μ {\displaystyle \mu } we have τ n ( φ ( X n ) − φ ( μ ) ) → φ μ ′ ( Z ) {\displaystyle \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 E {\displaystyle \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 mathematicsPages displaying short descriptions of redirect targets
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