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]
A map between Banach spaces and isHadamard-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.
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]
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Basic concepts |
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Derivatives |
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Measurability |
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Integrals |
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Functional calculus |
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