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In mathematics, the '''Hadamard derivative''' is a concept of [[Directional derivative|directional derivative]] for maps between [[Banach space]]s. It is particularly suited for applications in [[Stochastic programming|stochastic programming]] and [[Asymptotic theory (statistics)|asymptotic statistics]].<ref>{{Cite journal|last=Shapiro|first=Alexander|year=1990|title=On concepts of directional differentiability|journal=Journal of Optimization Theory and Applications|volume=66|issue=3|pages=477–487|doi=10.1007/bf00940933|citeseerx=10.1.1.298.9112|s2cid=120253580}}</ref>
== Definition ==
A map <math>\varphi : \mathbb{D}\to \mathbb{E}</math> between [[Banach space]]s <math>\mathbb{D}</math> and <math>\mathbb{E}</math> is '''Hadamard-directionally differentiable'''<ref name=":0">{{Cite journal|last=Shapiro|first=Alexander|year=1991|title=Asymptotic analysis of stochastic programs|journal=Annals of Operations Research|volume=30|issue=1|pages=169–186|doi=10.1007/bf02204815|s2cid=16157084}}</ref> 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>\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.<ref name=":0" />
* The Hadamard derivative is readily generalized for maps between [[Hausdorff space|Hausdorff]] [[Topological vector space|topological vector space]]s.

== Applications ==
A version of functional [[Delta method|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 model|econometric model]]s, including models with partial identification and weak instruments.<ref>{{Cite arXiv |last1=Fang|first1=Zheng|last2=Santos|first2=Andres|year=2014|title=Inference on directionally differentiable functions|eprint=1404.3763|class=math.ST}}</ref>

==See also==

* {{annotated link|Directional derivative}}
* {{annotated link|Fréchet derivative}} - generalization of the total derivative
* {{annotated link|Gateaux derivative}}
* {{annotated link|Generalizations of the derivative}}
* {{annotated link|Total derivative}}

==References==

{{reflist}}


[[Category:Directional statistics]]
[[Category:Generalizations of the derivative]]

{{Sourceattribution|Hadamard derivative}}