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]\displaystyle{ \varphi : \mathbb{D}\to \mathbb{E} }[/math] between Banach spaces [math]\displaystyle{ \mathbb{D} }[/math] and [math]\displaystyle{ \mathbb{E} }[/math] is Hadamard-directionally differentiable[2] at [math]\displaystyle{ \theta \in \mathbb{D} }[/math] in the direction [math]\displaystyle{ h \in \mathbb{D} }[/math] if there exists a map [math]\displaystyle{ \varphi_\theta': \, \mathbb{D} \to \mathbb{E} }[/math] such that[math]\displaystyle{ \frac{\varphi(\theta+t_n h_n)-\varphi(\theta)}{t_n} \to \varphi_\theta'(h) }[/math] for all sequences [math]\displaystyle{ h_n \to h }[/math] and [math]\displaystyle{ t_n \to 0 }[/math]. Note that this definition does not require continuity or linearity of the derivative with respect to the direction [math]\displaystyle{ 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]\displaystyle{ X_n }[/math] be a sequence of random elements in a Banach space [math]\displaystyle{ \mathbb{D} }[/math] (equipped with Borel sigma-field) such that weak convergence [math]\displaystyle{ \tau_n (X_n-\mu) \to Z }[/math] holds for some [math]\displaystyle{ \mu \in \mathbb{D} }[/math], some sequence of real numbers [math]\displaystyle{ \tau_n\to \infty }[/math] and some random element [math]\displaystyle{ Z \in \mathbb{D} }[/math] with values concentrated on a separable subset of [math]\displaystyle{ \mathbb{D} }[/math]. Then for a measurable map [math]\displaystyle{ \varphi: \mathbb{D}\to\mathbb{E} }[/math] that is Hadamard directionally differentiable at [math]\displaystyle{ \mu }[/math] we have [math]\displaystyle{ \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]\displaystyle{ \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
- 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
References
- ↑ Shapiro, Alexander (1990). "On concepts of directional differentiability". Journal of Optimization Theory and Applications 66 (3): 477–487. doi:10.1007/bf00940933.
- ↑ 2.0 2.1 Shapiro, Alexander (1991). "Asymptotic analysis of stochastic programs". Annals of Operations Research 30 (1): 169–186. doi:10.1007/bf02204815.
- ↑ Fang, Zheng; Santos, Andres (2014). "Inference on directionally differentiable functions". arXiv:1404.3763 [math.ST].
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