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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 between Banach spaces and is Hadamard-directionally differentiable[2] at in the direction if there exists a map such that for all sequences and .
Note that this definition does not require continuity or linearity of the derivative with respect to the direction . 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 be a sequence of random elements in a Banach space (equipped with Borel sigma-field) such that weak convergence holds for some , some sequence of real numbers and some random element with values concentrated on a separable subset of . Then for a measurable map that is Hadamard directionally differentiable at we have (where the weak convergence is with respect to Borel sigma-field on the Banach space ).
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. CiteSeerX 10.1.1.298.9112. doi:10.1007/bf00940933. S2CID 120253580.
- ^ a b Shapiro, Alexander (1991). "Asymptotic analysis of stochastic programs". Annals of Operations Research. 30 (1): 169–186. doi:10.1007/bf02204815. S2CID 16157084.
- ^ Fang, Zheng; Santos, Andres (2014). "Inference on directionally differentiable functions". arXiv:1404.3763 .
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