Английская Википедия: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.[1]
Definition
A map <math>\varphi : \mathbb{D}\to \mathbb{E}</math> between Banach spaces <math>\mathbb{D}</math> and <math>\mathbb{E}</math> is Hadamard-directionally differentiable[2] at <math>\theta \in \mathbb{D}</math> in the direction <math>h \in \mathbb{D}</math> if there exists a map <math>\varphi_\theta': \, \mathbb{D} \to \mathbb{E}</math> such that <math display=block>\frac{\varphi(\theta+t_n h_n)-\varphi(\theta)}{t_n} \to \varphi_\theta'(h)</math> for all sequences <math>h_n \to h</math> and <math>t_n \to 0</math>.
Note that this definition does not require continuity or linearity of the derivative with respect to the direction <math>h</math>. 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.[2]
- 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 <math>X_n</math> be a sequence of random elements in a Banach space <math>\mathbb{D}</math> (equipped with Borel sigma-field) such that weak convergence <math>\tau_n (X_n-\mu) \to Z</math> holds for some <math>\mu \in \mathbb{D}</math>, some sequence of real numbers <math>\tau_n\to \infty</math> and some random element <math>Z \in \mathbb{D}</math> with values concentrated on a separable subset of <math>\mathbb{D}</math>. Then for a measurable map <math>\varphi: \mathbb{D}\to\mathbb{E}</math> that is Hadamard directionally differentiable at <math>\mu</math> we have <math>\tau_n (\varphi(X_n)-\varphi(\mu)) \to \varphi_\mu'(Z)</math> (where the weak convergence is with respect to Borel sigma-field on the Banach space <math>\mathbb{E}</math>).
This result has applications in optimal inference for wide range of econometric models, including models with partial identification and weak instruments.[3]
See also
- Шаблон:Annotated link
- Шаблон:Annotated link - generalization of the total derivative
- Шаблон:Annotated link
- Шаблон:Annotated link
- Шаблон:Annotated link
References
Шаблон:Analysis in topological vector spaces
